The ordinates of the points P and Q on the parabola with focus $(3,0)$ and directrix $x=-3$ are in the ratio $3:1$. If $R(\alpha, \beta)$ is the point of intersection of the tangents to the parabola at P and Q, then $\frac{\beta^{2}}{\alpha}$ is equal to ______.

JEE Main · 2023 · 08 Apr 2023 · Shift 2 · Parabola

The ordinates of the points P and Q on the parabola with focus

(3, 0)

and directrix

x = - 3

are in the ratio

3 : 1

. If

R (α, β)

is the point of intersection of the tangents to the parabola at P and Q, then

\frac{β ^{2}}{α}

is equal to ______.

Answer: 16

Solution

The parabola with focus

(3, 0)

and directrix

x = - 3

has equation

y^{2} = 12 x

.

Let

P = (3 t_{1}^{2}, 6 t_{1})

and

Q = (3 t_{2}^{2}, 6 t_{2})

be points on the parabola. Since ordinates are in ratio

3 : 1

, we have

\frac{6 t _{1}}{6 t _{2}} = 3

, giving

t_{1} = 3 t_{2}

.

The point of intersection of tangents at P and Q is:

$α = 3 t_{1} t_{2} = 3 (3 t_{2}) t_{2} = 9 t_{2}^{2}$
$β = 3 (t_{1} + t_{2}) = 3 (3 t_{2} + t_{2}) = 12 t_{2}$

Therefore,

\frac{β ^{2}}{α} = \frac{144 t _{2}^{2}}{9 t _{2}^{2}} = 16

JEE Main · 2023 · 08 Apr 2023 · Shift 2 · Integer answer

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