EQUATION OF CHORD OF CONTACT OF TANGENTS FROM A  POINT

Find the equation to the chord of contact of tangents from the point

(i)  (− 3, 1) to the parabola y² = 8x

(ii)  (2, 4) to the ellipse  2x²+5y²  =  20

(iii)  (5, 3) to the hyperbolae  4x²-6y²  =  24

Question 1 :

(− 3, 1) to the parabola y²  =  8x

Solution :

To find equation of chord of contact to tangents, we should do the following changes in the equation of the curve.

x²  ==>  xx_1, y²  ==>  yy_1, x  =  (x+x₁)/2 and y  =  (y+y₁)/2

Equation of Chord of Contact of Tangents at (x₁, y₁)

 yy₁  =  8[(x+x₁)/2]

 yy₁  =  4(x+x₁)

The chord of contact of tangents is passing through the point (-3, 1).

 y(1)  =  4(x-3)

y = 4x-12

4x-y-12  =  0

Question 2 :

(2, 4) to the ellipse 2x² + 5y²  =  20

Solution :

x²  ==>  xx₁ and y²  ==>  yy₁

Equation of Chord of Contact of Tangents at (x₁, y₁)

2xx₁ + 5yy₁  =  20

The chord of contact of tangents is passing through the point (2, 4).

2x(2) + 5y(4)  =  20

4x+20y  =  20

x+5y  =  5

Question 3 :

 (5, 3) to the hyperbola 4x²-6y²  =  24

Solution :

x²  ==>  xx₁ and y²  ==>  yy₁

Equation of Chord of Contact of Tangents at (x₁, y₁)

4x²-6y²  =  24

The chord of contact of tangents is passing through the point (5, 3).

4x(5) - 6y(3)  =  24

20x-18y  =  24

10x-9y  =  12

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