TS Inter second year Maths 2B Concept

maths IIB feature image head

This note is designed by the ‘Basics in Maths’ team. These notes to do help the TS intermediate second-year Maths students fall in love with mathematics and overcome the fear.

These notes cover all the topics covered in the TS I.P.E second year maths 2B syllabus and include plenty of formulae and concept to help you solve all the types of Inter Math problems asked in the I.P.E and entrance examinations.


1. CIRCLES

Circle: In a plane, the set of points which are at a constant distance from a fixed point is called a circle.

circle for second year maths

∗ The fixed point is called centre (C) of the circle and constant distance is called radius(r) of the circle

Unit circle: If the radius of the circle is 1 unit, then that circle is called the unit circle.

unit circle

Point Circle: A circle is said to be a point circle if its radius is zero. A point circle contains only one point in the centre of the circle.  •

∗ The equation of the circle with centre (h, k) and radius r is            equation of the circle with centre origin and radius r

(x – h)2 + (y – k)2 = r2                 

∗ The equation of the circle with centre origin and radius r isequation of the circle with centre origin and radius r

x2 + y2 = r2

⇒ x2 + y2 = r2 is called standard form of the circle.

The general equation of the second degree ax2 + 2hxy + by2 + 2gx + 2fy + c = 0, where a, b, f, g, h and c are real numbers, represent a circle iff (i) a = b ≠ 0 (ii) h = 0 and (iii) g2 + f2 + c ≥ 0

∗ The general equation of the circle is x2 + y2 + 2gx + 2fy + c = 0

It’s centre c = (– g, – f) and radius TS inter circle radius 1

∗ The equation of the circle passing through origin is x2 + y2 + 2gx + 2fy = 0.

∗ The equation of the circle whose centre on the x-axis is x2 + y2 + 2gx + c = 0.

∗ The equation of the circle having centre on y-axis is x2 + y2 + 2fy + c = 0.

∗ The circles which have the same centre are called concentric circles.

∗ The equation of the circle concentric with the circle x2 + y2 + 2gx + 2fy + c = 0 is

x2 + y2 + 2gx + 2fy + k = 0.

∗ The length of the intercept made by a circle x2 + y2 + 2gx + 2fy + c = 0 onlength of the intercepts made by circle

  • x -axis is TS inter circle length of intercept by x axis  if g2 – c > 0
  • y -axis is TS inter circle length of intercept by y axisif f2 – c > 0

Note: –

(a) if g2 – c = 0, then A1 A2 = 0 ⇒ the circle touches the x- axis at only one point.

(b)  if f2 – c = 0, then B1 B2 = 0 ⇒ the circle touches the y- axis at only one point.

(c) if g2 – c < 0, then the circle does not meet the x- axis.

(d) if f2 – c < 0, then the circle does not meet the y- axis.

∗ The equation of the circle having the line segment joining A (x1, y1) and B (x2, y2) as a diameter is

equation of the circle passing through end points of diameter

(x – x1) (x – x2) + (y – y1) (y – y2) = 0.

 

∗ Let A, B be any two points on a circle then,secantline and chord of the circle

  • The line is called secant line of the circle.
  • The line segment is called chard of the circle.
  • AB is called the length of the chord.

∗ A chord passing through the centre is called the diameter of the circle.

∗ The angle subtended by a chord on the circumference of at any point is equal.

angle sustended by chord at any point

The perpendicular bisector of a chord of a circle is asses through the centre of the circle.

perpendicular bisector of the chord assing through centre

∗ The angle in a semicircle is 900.

angle in a semicircle

 

∗ The equation of the circle passing through three non-collinear points A (x1, y1), B (x2, y2), C (x3, y3) is

equatin of the circle passing through three points

Where ci = − (x2 + y2) and i = 1,2,3

∗ centre of the circle is

centre of the circle lassing through 3 points

Parametric form:

If P (x, y) is a point on the circle with centre (h, k) and radius r, then

X = h + r cosθ, y = k + r sinθ  0 ≤ θ ≤ 2π.

⇒ A point n the circle x2 + y2 = r2 is taken as (r cosθ, r sinθ) and simply denoted by θ.

      Note:

  1.  If the centre of the circle is the origin, then the parametric equations are x = r cosθ, y = r, 0 ≤ θ ≤ 2π.
  2. The point (h + rcosθ1, k + r sin θ1) is referred to as the point θ1 on the circle having the centre (h, k) and radius r.

Notations:

S = x2 + y2 + 2gx + 2fy + c

S1 = xx1 + yy1 + g(x +x1) + f (y +y1) + c

S11 = x12 + y12 + 2gx1 +2fy1 + c

S12 = x1x2 + y1y2 + g(x1 + x2 ) + f (y1 + y2) + c


Position of a point with respect to the circle: ts inter 2B position of point

A circle in a plane divides the plane into three parts.
1. The interior of the circle

2. The circumference which is the circular curve.

3. The exterior of the circle.

Power of point:

Les S = 0 be a circle with radius ‘r’ and centre ‘C’ and P (x1, y1) be a point on the circle, then CP – r2 is called the power of point ‘P’ concerning S = 0.

power of point

  • The power of point P (x1, y1) w.r.t. S = 0 is S11.

•Let S = 0 be a circle in a plane and P (x1, y1) be any point in the same plane. thents inter 2B position of the point 2

  1. P lies in the interior of the circle ⇔ S11 < 0.
  2. P lies on the circle ⇔ S11 = 0.
  3. Plies in the exterior of the circle ⇔ S11 > 0.

Secant and tangent of a circle:

Let P be any point on the circle and Q be neighbourhood point of P lying on the circle. join P and Q, then the line PQ is the secant line.ts inter 2B secant and tangent to a circle

The limiting position of the secant line PQ when Q is approached to the point P along the circle is called a tangent to the circle at P.

Length of the tangent:ts inter 2B length of tangent

If P is any point on the circle S = 0 and T is any exterior point of the circle, then PT is called the length of the tangent.

∗ If S = 0 is a circle and P (x1, y1) is an exterior point with respect o S = 0, then the length of the tangent from P (x1, y1) to S =0 is TS inter circle length of tangent

Condition for a line to be a tangent:

  • A straight-line y = mx + c (i) meet the circle x2 + y 2 = r2 in two distinct points if ts inter 2B condition for a line to be a tangent 1
  • Touch the circle x2 + y 2 = rif ts inter 2B condition for a line to be a tangent 2
  • Does not touch the circle x2 + y 2 = r2 in two distinct points if ts inter 2B condition for a line to be a tangent 3

Note:

  1. For all real values of m, the straight line ts inter 2B condition for a line to be a tangent 4 is a tangent to the circle x2 + y2 = r2 and the slope of the line is m.
  2. A straight-line y = mx + c is a tangent to the circle x2 + y2 = r2 if c = ts inter 2B condition for a line to be a tangent 5 .
  3. The equation of a tangent to the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 having the slope m is  ts inter 2B condition for a line to be a tangent 6where r is the radius of the circle.
  4. The circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 touches (i) x – axis if g2 = c (ii) y – axis if f2= c.

Chord joining two points on a circle:

If P (x1, y1) and Q (x2, y2) are two points on the circle S = 0 then the equation of secant line PQ is S1 + S2 = S12.

  Equation of tangent at a point on the circle:

The equation of the tangent at the point (x1, y1) to the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 is S1 = 0.

The equation of the tangent at the point (x1, y1) to the circle x2 + y2 = r2 is xx1 + yy1 – r2 = 0.

Point of contact:

If a straight-line lx + my + n = 0 touches the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 at P (x1, y1), then this line is the tangent to the circle S = 0. And the equation of the tangent is

     (x1 + g) x + (y1 +f) y + (gx1 + fy1 + c) = 0.

∗ The equation of the chord joining two points θ1, θ2 on the circle x2 + y2 + 2gx + 2fy + c = 0 is

ts inter 2B equation of chord in parametric form1

∗ The equation of the chord joining the points θ1, θ2 on the circle x2 + y2 = r2 is

ts inter 2B equation of chord in parametric form2

∗ The equation of the tangent at P(θ) on the circle x2 + y2 + 2gx + 2fy + c = 0 is

ts inter 2B equation of tangent in parametric form

∗ The equation of the tangent at P(θ) on the circle x2 + y2 = r2 is

        x cosθ + y sinθ = r.

Normal:ts inter 2B normal to the circle

The normal at any point P of the circle is the line which is passing through P and is perpendicular to the tangent at P.

 

  • The equation of the normal at P (x1, y1) of the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 is

                                             (x – x1) (y1 + g) – (y – y1) (x1 + g) = 0.

  • The equation of the normal at P (x1, y1) of the circle x2 + y 2 = r2 is xy1 – yx1 = 0.

Chord of contact and Polar:

∗ If P (x1, y1) is an external point of the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0, then there exists two tangents from P to the circle S = 0.ts inter 2B angents from external point to the circle

 

 

 

 

Chord of contact: –

If the tangents are drawn through P (x1, y1)ts inter 2B chord of contact diagram

 to a circle S = 0 touch the circle at points A and B then the secant line AB is called the chord of contact of P with respect to S = 0

∗ If P (x1, y1) is an external point of the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0, then the equation of the chord of contact of P with respect to S =0 is S1 = 0.

Note:

  1. If the point P (x1, y1) is on the circle S = 0, then tangent itself can be defined as the chord of contact.
  2. If the point P (x1, y1) s an interior point of the circle S = 0, then the chord of contact does not exist.

Pole and Polar: –ts inter 2B pole and polar of circle

Let S = 0 be a circle and P be any point if any line drawn through the point Pin the plane other than the centre of S = 0.then the points of intersection meets the circle in two points A and B, of tangents drawn at A and B lie on a line called polar of P and P, is called Pole of polar.

∗ The equation of the polar of P (x1, y1) with respect to the circle S = 0 is S1 = 0.

Note: –

  1. If Plies outside the circle S = 0, then polar of P meets the circle in two points and the polar becomes the chord f contact of P.
  2. If P lies on the circle S = 0, then the polar P becomes the tangent at P o the circle.
  3. If P lies inside the circle S = 0, then the polar of P does not meet the circle.
  4. If P is the centre of the circle S = 0, then the polar of P does not exist.
  5. The pole of the line lx + my + n = 0 with respect to the circle x2 + y2 = r2 is
  6. The pole of the line lx + my + n = 0 with respect to the circle x2 + y2 + 2gx + 2fy + c = 0 is
  7. The polar of P (x1, y1) with respect to the circle S = 0 passes through Q (x2, y2)  ⟺ polar of Q passes through P.

Conjugate points: Two P and Q are said to be conjugate points with respect to the circle S = 0, if polar of P with respect to S = 0 passes through Q.

 ⇒ The condition for the points P (x1, y1), Q (x2, y2) to be conjugate with respect to the circle S = 0 is S12 = 0.

Conjugate lines: Two lines L1 = 0 and L2 = 0 are said to conjugate lines with respect to the circle S = 0 if the pole of L1 = 0 is lies on L2 = 0.

The condition for the lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 to be conjugate with respect to the circle S ≡ x2 + y2 + 2gx + 2fy + c = 0 is

                                                r2 (l1l2 + m1m2) = (l1g + m1f – n1) (l2g + m2f – n2)

⟹ The condition for the lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 to be conjugate with respect to the circle S ≡ x2 + y2 = r2 is  r2 (l1l2 + m1m2) = n1n2

Inverse points: Let S = 0 be a circle with centre C and radius r. two points P and Q are said to be inverse points with respect to the circle S = 0 if

  1. C, P, Q are collinear.
  2. P, Q lies on the same side of C.
  3. CQ = r2.

⟹  If lies inside of the circle S = 0, then Q lies outside of the circle.

⟹  If P lies on the circle S = 0, then P =Q.

⟹  Let S = 0 be a circle with centre C and radius r. The polar of P meets the line CP in Q iff  P, Q are inverse points.

⟹  f P, Q are inverse points with respect to S = 0, then P, Q are conjugate points with respect to the circle S = 0.

⟹  If P, Q are inverse points with respect to S = 0, then Q is the foot of the perpendicular from P on the polar of P with respect to the circle S = 0.

Equation of the chord with the given middle point:

The equation of the chord of the circle S = 0 having P (x1, y1) as its midpoint is S1 = S11.

Common tangents to the circle:ts inter maths 2B common tangent to the circle

 ⟹ A straight line L = 0 is said to be common tangent to the circle S = 0 and S= 0 if it is a tangent to both S = 0 and S’ = 0.                       

Two circles are said to be touch each other if they have only one common tangent.

The relative position of two circles:

Let C1, C2 centres and r1, r2 be the radii of two circles S = 0 and S’ = 0respectively.

1.If C1C2 > r1+ r2, then two circles do not intersect.ts inter maths 2B does not meet the circles

⟹2 direct common tangents and

 2 transverse common tangents

  Total 4 common tangents

 ⟹P divides C1C2 in the ratio r1: r2 internally   

     Here P is called the internal centre of similitude (I.C.S)

  ⟹ Q divides C1C2 in the ratio r1: r2 externally   

     Here Q is called the external centre of similitude (E.C.S)               

2.If C1C2 = r1+ r2, two circles touch each otherts inter maths 2B the circles touch each other externally

   ⟹ Q divides C1C2 in the ratio r1: r2 externally  

⟹ two direct common tangents and one common tangent. Total 3 tangents

  Here Q is called the external centre of similitude (E.C.S)

3. ts inter maths 2B the circles intersects at distinct points1

⟹ two direct common tangentsts inter maths 2B the circles intersects at distinct points

    Here Q is called the external centre of similitude (E.C.S)
   ⟹
Q divides C1C2 in the ratio r1: r2 externally   

⟹ internal centre of similitude does not exist.

4. ts inter maths 2B the circles touch internally1ts inter maths 2B the circles touch internally

⟹ only one common tangent

internal centre of similitude does not exist.

 

5. ts inter maths 2B one circle lies inside the other circle1     ts inter maths 2B one circle liec inside the other circle

no. of common tangents zero.

 

 Note: the combined equation of the pair of tangents drawn from an external point P (x1, y1) to the circle S = 0 is S S11 = S12.

 


2.SYSTEM OF CIRCLES

A set of circles is said to be a system of circles if it contains at least two circles.

The angle between two intersecting circles:ts inter 2B angle between two circles diagram

If two circles S = 0 and S’ = 0 intersect at P then the angle between the tangents of two circles at P is called angle between the circles at P.

⟹ If two circles S = 0 and S’ = 0 intersect at P and Q then the angle between the tangents of two circles at P and Q are equal.

⟹ If d is the distance between the centres of the two intersecting circles with radii r1, r2 and θ is the angle between the circles then.

ts inter 2B angle between two circles when d r1 r2 given

⟹ If θ is the angle between the circles x2 + y2 + 2gx + 2fy + c = 0 and x2 + y2 + 2g’x + 2f’y + c’ = 0 then

ts inter 2B angle between two circles when gg' ff' cc' known

⟹ Two intersecting lines are said to be Orthogonal if the angle between the circles is a right angle.

Condition for the orthogonality:

⟹ The condition that the two circles x2 + y2 + 2gx + 2fy + c = 0 and x2 + y2 + 2g’x + 2f’y + c’ = 0 cut each other orthogonally is 2gg’ + 2ff’ = c + c’.

  ⟹ If d is the distance between the centres of the two intersecting circles with radii r1, r2. Two circles cut orthogonally if d2 = r12 + r22.

∎  If S = 0, S’ = 0 are two circles intersecting at two distinct points, then S – S’ = 0 represents a common chord of these two circles.

∎ If S = 0, S’ = 0 are two circles touch each other, then S – S’ = 0 represents a common tangent of these two circles.

∎ If S ≡ x2 + y2 + 2gx + 2fy + c =  0 and L ≡ lx + my + n = 0 are the equation of the circle and  a line respectively intersecting each other, then S + λ L = 0 represent a circle passing through the point intersection of  S = 0 and L = 0 ∀ λ ∈ ℛ.

Radical axes:ts inter 2B radical axis diagram

The radical axis of two circles s defined as the locus of the point which moves so that its powers with respect to the two circles are equal.

(OR)
The locus of a point, for which the powers with respect to given non-concentric circles are equal, is a straight line is called Radical axis of the given circles.

∎ The equation of Radical axis f the circles S = 0 and S’ = 0 is S – S’ = 0.

The radical axis of any two circles is perpendicular to the line joining their centres.

The lengths of tangents from a point on the radical axis of two circles are equal if exist.

Radical axis of two circles bisects all common tangents of the two circles.

∎ If the centres of any three circles are non-collinear then the radical axis of each pair of circles chosen from these three circles re concurrent.   

ts inter 2B radical axes of three circles are concurrent

Radical centre:  The point of concurrence of radical axes of each pair of three circles is called radical centre (see above figure).

∎ If the circle S = 0 cuts the each of the two circle S’ = 0 and S’’ =0 orthogonally then the centre of S =0 lies on the radical axis of S’ = 0 and S’’ = 0.

∎ Radical axis of two circles is

  • The ’common chord’ if the two circles intersect at two distinct points.
  • The ‘common tangent’ at the point of contact if the two circles touch each other.

The radical axis of any two circles bisects the line joining the points of contact of common tangents to the circles.

Let S = 0, S’ = 0 and S’’ =0 be three circles whose centres are non- collinear and no two circles of these are intersecting then the circles having

  • Radical centre of these circles as the centre of the circle.
  • Length of the tangent from the radical centre to any one of these three circles cuts the given three circles orthogonally.

CONIC SECTIONSts inter 2B conic sections diagram

Conic: The locus of a point moving on a plane such that its distance from a fixed point and a fixed straight line is in the constant ratio is called Conic.

OR

The locus of a point moving on a plane such that its distance from a fixed point and a fixed line on the plane are in a constant ratio ‘e’, is called a Conic.

Focus: The fixed point is called focus and it is denoted by S.

Directrix: The fixed straight line is called the directrix.

Eccentricity: The constant ratio is called eccentricity and it is denoted by ‘e’.

Conic is the locus of a point P moving on a plane such that SP/PM = e, PM is the perpendicular distance from P to directrix at M.

If e = 1, then the conic is parabola.

if 0 < e < 1, then the conic is Ellipse.

if e > 1, then the conic is Hyperbola.


3.PARABOLA

 

If e = 1, then the conic is parabola.TS inter 2B parabola diagram

  • The standard form of parabola is y2 = 4ax.
  • Focus S = (a. 0).
  • Equation of directrix is x + a = 0.
  • Vertex A = (0, 0) and A is the mid-point of SZ.

• Equation of the parabola with focus (α, β) and directrix lx + my + n = 0 is

ts inter 2B equation of the parabola 1

  • If the focus is situated on the left side of the directrix, the equation of the parabola with vertex as the origin and the axis is X-axis is y2 = – 4ax.
  • The vertex being the origin, if the axis of the parabola is taken as Y – axis, equation of the parabola is x2 = 4 ay or x2 = – 4 ay according to the focus is above or below the X-axis.

Nature of the curve:

The nature of the parabola f the equation y2 = 4 ax (a>0)

  • F y = 0, then 4 ax = 0 and x = 0

∴ the curve passes through the origin.

  • If x = 0, then y2 = 0. Which gives y = 0. Y – axis is the tangent to the parabola at origin.
  • Let P(x, y) be any point on the parabola (a>0) and y2 = 4 ax, we have x ≥ 0 and ts inter 2B equation of the parabola 2 

∴ for any positive real value of x, we obtain two value of y of equal magnitude but opposite in sign. This shows that the curve is symmetric about X-axis and lies in the first and fourth quadrants.

The curve does not exist on the left side of the Y-axis since x ≥ 0 for any point (x, y) on the parabola.

Chord: The line segment joining two points on a parabola is called a chord.

Focal chord: A chord which is passing through focus is called Focal Chord.

Double ordinate: A chord through a point P on the parabola, which is perpendicular to the axis of the parabola is called Double ordinate.

Latus rectum: The double ordinate passing through the focus is called Latus rectum.

⟹ Length of Latus rectum = 4a.

Various forms of the parabola

1. y2 = 4axTS inter 2B parabola diagram1

    focus: (a, 0)

    equation of directrix: x + a = 0

    axis of parabola: y = 0

    vertex: (0. 0)

2. y2 = −4axTS inter 2B parabola diagram2

    equation of directrix: x − a = 0                                 

  focus: (−a, 0)

    axis of parabola: y = 0

    vertex: (0. 0)

3. x2 = 4ayTS inter 2B parabola diagram3

    focus: (0, a)

    equation of directrix: y + a = 0

    axis of parabola: x = 0

    vertex: (0. 0)

4. x2 = −4axTS inter 2B parabola diagram4

    focus: (0, −a)

    equation of directrix: y − a = 0

    vertex:  (0. 0)    axis of parabola: x = 0

5. (y – k) 2 = 4a (x – h)TS inter 2B parabola diagram5

    focus: (h + a, k)

    equation of directrix: x – h + a = 0

    axis of parabola: y – k = 0

    vertex: (h. k)

6. (y – k) 2 = −4a (x – h)TS inter 2B parabola diagram6

  focus: (h – a, k)
   equation of directrix: x – h – a = 0
 

    axis of parabola: y – k = 0

    vertex: (h. k)

7. (x – h) 2 = 4a (y – k)TS inter 2B parabola diagram7

    focus: (h, k + a)

    equation of directrix: y – k + a = 0

    axis of parabola: x – h = 0

  vertex: (h. k)

8. (x – h)2 = −4a (y – k) TS inter 2B parabola diagram8

    focus: (h, k – a)

    equation of directrix: y – k – a = 0

    vertex: (h. k)
    axis of parabola: x – h =0

9. TS inter 2B parabola equation focus is in quadrantTS inter 2B parabola diagram9

    focus: (α, β)

    equation of directrix: lx + my + n = 0

    axis of parabola: m (x – α) – l (y – β) = 0

    vertex: A 

Note:  

  1. Equation of the parabola whose axis parallel to X – axis is x = ly2 + my + n.
  2. Equation of the parabola whose axis parallel to Y – axis is y = lx2 + mx + n.

Focal distance:  The distance of a point on the parabola from its focus is called Focal distance.

⟹ Focal distance of parabola s x1 + a

Parametric equations of a parabola:

The point (at2, 2at) satisfy the equation y2 = 4ax, the parametric equations of parabola are  x = at2, y = 2at. The point P(at2, 2at) is generally denoted by the point ‘t’ or P(t).

Notation:

  1. S ≡ y2 – 4 ax
  2. S1 ≡yy1 – 2a (x + x1)
  3. S12 ≡ y1y2 – 2a (x1 + x2)
  4. S11 ≡ y12 – 4 ax1

Equation of a tangent and normal at a point on the parabola:

∎ y = mx + c is a tangent to the parabola y2 = 4ax, then c = a/m or a =cm, and the point of contact is (a/m2, 2a/m).

∎ if m = 0, the line y = c is parallel to the axis of the parabola (i.e., x – axis)

    y = c ⟹ c2 = 4ax ⟹ x = c2 /4a

  ∴ point of contact is (c2/4a, c).

∎ if m ≠ 0 and c = 0, then

   Y = mx ⟹ x = 4a/m2 and y = 4a/m

 ∴ point of contact is (4a/m2, 4a/m).

∎ The equation of the chord joining the points (x1, y1) and (x2, y2) is S1 + S2 = S12.

∎ The equation of the tangent at P (x1, y1) to the parabola S = 0 is S1= 0.

∎ The equation of the normal at P (x1, y1) is (y – y1) = – y1/2a (x – x1).

Parametric form:

∎ The equation of the tangent at a point ‘t’ on the parabola y2 = 4ax is x – yt + at2 = 0.

∎ Equation of the normal at a point ‘t’ on the parabola y2 = 4ax is y + xt = 2at + at3.

∎ The condition for the straight-line lx + my + n = 0 to be a tangent to the parabola y2 = 4 ax is

      am2 = nl and point of contact is (n/l, –2 am/l).

∎ common tangent to the parabolas y2 = 4 ax and x2 = 4 by is x a1/3 + y b1/3 + a2/3 b2/3 = 0.

∎ The equation of the chord of contact of the external point (x1, y1) w.r. t parabola S = 0 is S1 = 0.

∎ The equation of the polar of the point (x1, y1) w.r. t parabola S = 0 is S1 = 0.

∎ The pole of the line lx + my + n = 0 w.r.t. parabola y2 = 4ax is (n/l, -2am/l).

∎ If two points P (x1, y1), Q (x2, y2) are conjugate points w.r.t. parabola S = 0, then S12 = 0.

∎ The lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 are conjugate lines with respect to the parabola y2 = 4 ax, then l1n2 + l2n1 = 2a m1m2.


4.ELLIPSE

Ellipse: A conic with eccentricity less than unity s called Ellipse.TS inter 2B ellipse diagram

∎ Equation of Ellipse in standard form is TS inter 2B standard form of ellipse equation

⇒b2 = a2 (1 – e2) ⇒ e2 – 1 = -b2/a2

TS inter 2B standard form of ellipse equation2

Major and Minor axis:

⟹ The line segment AA’ and BB’ of length 2a and 2b respectively are axes of the Ellipse.

⟹ If a > b AA’ is called Major axis and BB’ is called Minor axis and vice-versa if a<b.

Various form of Ellipse:

1.TS inter 2B ellipse form equation1TS inter 2B ellipse diagram1

Major axis: along the x-axis

Length of Major axis:2a

Minor axis: along y – axis

Length of Minor Axis:2b

Centre: (0, 0)

Foci: S = (ae, 0) and S’ = (–ae, 0)

Equation of directrices: x = a/e and x = –a/e

Eccentricty: TS inter 2B eccentricity of ellipse form equation1

2.TS inter 2B ellipse form equation2     TS inter 2B ellipse diagram2

Major axis: along y – axis

Length of Major axis:2b

Minor axis: along x – axis

Length of Minor Axis:2a

Centre: (0, 0)

Foci: S = (0, be) and S’ = (0, –be)

Equation of directrices: x = b/e and x = –b/e

Eccentricty:TS inter 2B eccentricity of ellipse form equation2

Centre not at the origin

3.TS inter 2B ellipse form equation3TS inter 2B ellipse diagram3

Major axis: along with y = k

Length of Major axis:2a

Minor axis: along x = h

Length of Minor Axis:2b

Centre: (h, k)

Foci: S = (h +ae, k) and S’ = (h – ae, k)

Equation of directrices: x = h + a/e and x = h – a/e

Eccentricty:TS inter 2B eccentricity of ellipse form equation3

4.TS inter 2B eccentricity of ellipse form equation4TS inter 2B ellipse diagram4

Major axis: along x = h

Length of Major axis:2b

Minor axis: along with y = k

Length of Minor Axis:2a

Centre: (h, k)

Foci: S = (h, k + be) and S’ = (h, k – be)

Equation of directrices: xy = k + b/e and y = k – b/e

Eccentricty: TS inter 2B eccentricity of ellipse form equation4

Chord: The line segment joining two points on a parabola is called a  chord of Ellipse.

Focal chord: A chord which is passing through one of the foci is called Focal Chord.

Latus rectum: A focal chord perpendicular to the major axis of the Ellipse is called Latus Rectum. Ellipse has two latera recta.

 Length of the Latus rectum:

1.The coordinates of the four ends of the latera recta of the ellipse TS inter 2B eccentricity of ellipse form equation5

L = (ae, b2/a), L’ = (ae, -b2/a) and L1 = (-ae, b2/a), L1’= (-ae, -b2/a).

length of the Latus rectum = 2b2/a.

2.length of the Latus rectum of an ellipse  TS inter 2B eccentricity of ellipse form equation6 is 2a2/b and the coordinates of the four ends of the latera recta are

L = (a2/b, be), L’ = (-a2/b, be) and L1 = (a2/b, -be), L1’ = (-a2/b, -be).

3. The equation of the Latus rectum of the Ellipse  TS inter 2B eccentricity of ellipse form equation5 through S is x = ae and through S’ is x = -ae.

4. The equation of the Latus rectum of the Ellipse  TS inter 2B eccentricity of ellipse form equation6 through S is y = be and through S’ is y = -be.

5. If P (x, y) is any point on the Ellipse TS inter 2B ellipse equation without condition  whose foci are S and S’, then SP +S’P is constant.

Auxiliary circle: The circle described on the major axis of an Ellipse as the diameter is called the Auxiliary circle of the Ellipse. The Auxiliary circle of the Ellipse TS inter 2B eccentricity of ellipse form equation5 is x2 + y2 = a2.

 Parametric equations:  The parametric equations of the Ellipse TS inter 2B ellipse equation without condition are x = a cosθ and y = b sinθ.

Notation:

  TS inter 2B ellipse notation

Equation of Tangent and Normal

 The equation of any tangent to the Ellipse can be written as TS inter 2B quation of the tangent to ellipse

 The condition for a straight-line y = mx + c to be a tangent to the Ellipse    TS inter 2B ellipse equation without condition   is  c2 = am2 + b2.

∎ The point of contact of two parallel tangents to the Ellipse are (-a2m/c, b2/c) and (a2m/c, -b2/c)

∎ The equation of the chord joining two points (x1, y1) and (x2, y2) on the Ellipse S = 0 is S1 + S2 = S12.

∎ The equation of the Normal at P (x1, y1) to the Ellipse is TS inter2B equation of the normal to the ellipse

∎ Equation of the tangent at P(θ) on the Ellipse   TS inter2B equation of the tangentto the ellipse2 

∎ Equation of the normal at P(θ) on the Ellipse S = 0 isTS inter2B equation of the normal to the ellipse2

∎ When θ = 0, π; equation of Normal is y =0.

∎ When θ = π/2, 3π/2; equation of Normal is x =0.

∎ The condition for the line lx + my + n = 0 to be a tangent the Ellipse S = 0 is a2l2 + b2m2 = n2.

∎ The condition for the line x cosα + y sinα = p to be a tangent the Ellipse S = 0 is a2 cos2 α + b2 sin2 α = p2.

∎ The pole of the line lx + my + n = 0 with respect to the Ellipse S = 0 is (-a2l/n, -b2m/n).

∎ The condition for the two lines l1x + m1y + n1 = 0 and l2x + m2y + n2 = 0 to be conjugate with respect to the Ellipse S = 0 is a2l1l2 + b2m1m2 = n1n2. a2l1l2.


5.HYPERBOLA

Hyperbola: Hyperbola is a conic in which the eccentricity is greater than the unity.TS inter 2B Hyperbola diagram
Standard form of Hyperbola:

Equation of Hyperbola in standard form is TS inter 2B equation of standard form of Hyperbola

 Centre: C (0, 0)

Foci: (± ae, 0)

Directrix: x = ± a/e.

Ecdntricity: TS inter 2B eccentricity of Hyperbola

Notation:    TS inter 2B Hyperbola notation

Rectangular Hyperbola:

If in a Hyperbola the length of the transverse axis (2a) is equal to the length of the conjugate axis(2b), then the hyperbola is called rectangular hyperbola.

Its equation is x2 – y2 = a2 and eccentricity is sqrt 2 .

Auxiliary circle: The circle described on the transverse axis of hyperbola as diameter is called the auxiliary circle of the hyperbola.

The equation of the auxiliary circle of S = 0 is x2 + y2 = a2.

Parametric equations:  The parametric equations of the Parabola S = 0 are x = a secθ and y = b tanθ.

Conjugate Hyperbola:

The hyperbola whose transverse and conjugate axis are respectively the conjugate and transverse axis of a given hyperbola is called a conjugate hyperbola.

The equation of hyperbola conjugate to   S ≡  is S’ ≡

∎ For TS inter 2B equation of Hyperbola2

The transverse axis lies on along X-axis and its length is 2a.

The conjugate axis lies on along Y-axis and its length is 2b.
∎ For  TS inter 2B equation of conjugate Hyperbola

 The transverse axis lies on along Y-axis and its length is 2b.

The conjugate axis lies on along X-axis and its length is 2a.

Various form of Hyperbola:TS inter 2B various forms of Hyperbola diagram

Let TS inter 2B equation of Hyperbola2 and      TS inter 2B equation of conjugate Hyperbola

1.Hyperbola TS inter 2B equation of Hyperbola2

The transverse axis along X-axis: y =0

Length of the transverse axis:2a

The conjugate axis along Y-axis: x = 0

Length of the conjugate axis: 2b

Centre: (0, 0)

Foci: (± ae, 0)

Equation of the directrices: x = ± a/e

Eccentricity:TS inter 2B eccentricity of Hyperbola

2.Conjugate Hyperbola TS inter 2B equation of conjugate Hyperbola

The transverse axis along Y-axis: x = 0

Length of the transverse axis:2b

The conjugate axis along X-axis: y = 0

Length of the conjugate axis: 2a

Centre: (0, 0)

Foci: (0, ± be)

Equation of the directrices: y = ± b/e

Eccentricity:TS inter 2B eccentricity of conjugate Hyperbola

Centre not at the origin:

3.Hyperbola TS inter 2B equation of standard form of Hyperbola1

The transverse axis along X-axis: y = k

Length of the transverse axis:2a

The conjugate axis along Y-axis: x = h

Length of the conjugate axis:2b

Centre: (h, k)

Foci: (h± ae, k)            

Equation of the directrices: x = h± a/e

Eccentricity:TS inter 2B eccentricity of Hyperbola

4.Hyperbola TS inter 2B equation of conjugate Hyperbola2

The transverse axis along Y-axis: x = h

Length of the transverse axis:2b

The conjugate axis along X-axis: y = k

Length of the conjugate axis: 2a

Centre: (h, k)

Foci: (h, k ± be)

Equation of the directrices: y = k± b/e

Eccentricity: TS inter 2B eccentricity of conjugate Hyperbola

Equation of tangent and normal at a point on the Hyperbola:

∎ The equation of the tangent at P (x1, y1) to the hyperbola S = 0 is S1 = 0.

∎ The equation of the tangent at P(θ) on the hyperbola S = 0 is TS inter 2B equation of tangent to the Hyperbola

∎ The equation of the Normal at P (x1, y1) to the hyperbola S = 0 is TS inter 2B equation of normal to the Hyperbola

∎ Equation of the normal at P(θ) on the Hyperbola S = 0 is TS inter 2B equation of normal to the Hyperbola2

∎ The condition for a straight-line y = mx + c to be a tangent to the hyperbola S = 0 is c2 = am2 − b2.

Asymptotes of a hyperbola:

The equations of asymptotes of hyperbola S = 0 are    TS inter 2B asymptotes of Hyperbola  and the joint equation of asymptotes is TS inter 2B asymptotes of Hyperbola2


6. INTEGRATION

∎ Let E be a subset of R such that E contains a right or left the neighbourhood of each of its points and let f: E → R be a function. If there is a function F on E such that F’(x) = f(x) ∀ x ∈ E, then we call F an anti-derivative of f or a primitive of f.

Indefinite integral: Let f: I→R. Suppose that f has an antiderivative F on I. Then we say that f has an integral on I and for any real constant c, we call F + c an indefinite integral of over I, denote it by    ∫f(x) dx and read it as ‘integral’ f(x) dx.

∫f = ∫f(x) dx = F(x) + c. here c is called constant of integration.

In the indefinite integral ∫f(x)dx, f is called ‘integrand’ and x is called the variable of integration.

⟹  TS inte 2B integration formula 1

⟹ if f: I⟶R is differentiable on I, then ∫f’(x)dx = f(x) + c.

Standard forms:               

TS inte 2B integration standard forms

Properties of integrals:

∎ ∫ (f ±g) (x) dx = ∫f(x) dx ± ∫g(x) dx + c

∎ ∫(af) (x) dx = a ∫f(x) dx + c

∎ ∫ (f1 + f2 + … + fn) (x) dx = ∫f1(x) dx + ∫f2(x) dx + … +∫fn(x) dx +c

∎ ∫f(g(x)) g’(x) dx = F(g(x)) + c

 ∎ ∫f(ax + b) dx = 1/a F(ax +b) + c

Some important formulae:

  1. ∫eax dx = 1/a eax + c
  2. ∫sin (ax + b) dx = -1/a cos (ax + b) + c
  3. ∫cos (ax + b) dx = 1/a sin (ax + b) + c
  4. ∫sec2 (ax + b) dx = 1/a tan (ax + b) + c
  5. ∫ cossec2 (ax + b) dx = 1/a cot (ax + b) + c
  6. ∫cosec (ax + b) cot (ax + b) dx = -1/a cosec (ax + b) + c
  7. ∫sec (ax + b) tan (ax + b) dx = 1/a sec (ax + b) + c

TS inte 2B integration formula 3

TS inte 2B integration formula 4

Integration by parts:

Let u, v real valued differentiable functions in I. Suppose that u’v has an integral on I, then uv’ has an integral on I and

∫(uv’) (x) dx = (uv) – ∫(u’v) (x) dx + c or ∫(uv) dx = u ∫v dx – ∫ [u’ ∫v dx] dx + c

Integration of exponential functions:

∫ex dx = ex + c; ∫x ex dx = (x – 1) ex + c

∫ ex [f(x) +f’(x)] dx = ex f(x) + c

Integration of logarithmic functions:

∫log x dx = x log x – x + c

Integration of inverse trigonometric functions:

TS inte 2B integration of inverse trigonometric functions

Evaluation of integrals form  : TS inte 2B evaluation of integration form1

Working rule:  reduce ax2 + bx + c to the form of a[(x + α)2 + β] and then integrate using the substitution t = x + α.

 Evaluation of integrals form TS inte 2B evaluation of integration form2

Working rule:  

Case(i) if a >0 and b2 – 4ac < 0, then reduce ax2 + bx + c to the form of a[(x + α)2 + β] and then integrate.

Case(ii) if a <0 and b2 – 4ac >0, then reduce ax2 + bx + c to the form of (-a) [ β – (x + α)2 +] and then integrate.

Evaluation of integrals form TS inte 2B evaluation of integration form3

Working rule:  write px + q in the form of A (ax2 + bx +c)’ + B, then integrate.

Evaluation of integrals form  TS inte 2B evaluation of integration form4

Working rule:  write cos x =cos2(x/2) – sin2(x/2) and sin x = 2 sin(x/2) cos (x/2)

Put t = tan(x/2), then dt = ½ sec2 (x/2) dx

Cos x = 1 – t2 / 1 + t2, sin x = 2t/1 + t2 then integrate.

Evaluation of integrals form   TS inte 2B evaluation of integration form5