If one end of a focal chord $AB$ of the parabola $y^{2}=8 x$ is at $A\left(\frac{1}{2},-2\right)$, then the equation of the tangent to it at $B$ is:

JEE Main · 2020 · 09 Jan 2020 · Shift 2 · Parabola

If one end of a focal chord

A B

of the parabola

y^{2} = 8 x

is at

A (\frac{1}{2}, - 2)

, then the equation of the tangent to it at

B

is:

(A) $2 x - y - 24 = 0$
(B) $x - 2 y + 8 = 0$ ✓ Correct
(C) $2 x + y - 24 = 0$
(D) $x + 2 y + 8 = 0$

Answer: (B)

Solution

For

y^{2} = 8 x

4 a = 8

a = 2

, and a general point is

(2 t^{2}, 4 t)

. From

A (\frac{1}{2}, - 2)

4 t_{1} = - 2

gives

t_{1} = - \frac{1}{2}

. The endpoints of a focal chord satisfy

t_{1} t_{2} = - 1

, so

t_{2} = 2

, placing

B

at parameter

2

. The tangent at parameter

t

t y = x + a t^{2}

, which at

t = 2

becomes

2 y = x + 8

, that is

x - 2 y + 8 = 0

JEE Main · 2020 · 09 Jan 2020 · Shift 2 · Single correct option

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