Question

If tangents to the parabola $y^2=4x$ intersect the hyperbola at A & B, then find the locus of point of intersection of tangent at A and B.

• A. 4y² + 81x = 0
• B. y² + 81x = 0
• C. 4y² + 16x = 0
• D. y² + 16x = 0

Answer 1

The correct option is A

4y² + 81x = 0

Let p(h,k) be the point of intersection of tangents at A &B.

Equation of chord of contact AB is T=0

$\frac{hx}{4}-\frac{ky}{9}-1=0$.....(1)

The equation of chord of contact touches parabola

Equation of tangent to parabola$y^2=4x$

$y=mx+\frac{a}{m}$

$\Rightarrow y=mx+\frac{1}{m}$

$mx-y+\frac{1}{m}$.........(2)

Equation (1) and (2) must be the equation of same line there coefficient must be in the same ratio.

$\frac{\frac{h}{4}}{m}=\frac{\frac{-k}{9}}{-1}=\frac{-1}{\frac{1}{m}}$

$\underset{}{\frac{h}{4m}=}\underset{}{\frac{k}{9}=-m}$

$m=-\frac{k}{9}$

$\frac{h}{4m}=\frac{k}{9}$

substituting $m=\frac{-k}{9}$

$\frac{h}{\frac{4(-k)}{9}}=\frac{k}{9}$