First slide

Parabola

Question

The slope of the line touching both the parabolas
$y^{2} = 4 x$ and $x^{2} = - 32 y$ is

Moderate

A

$\frac{1}{2}$
B

$\frac{3}{2}$
C

$\frac{1}{8}$
D

$\frac{2}{3}$

Solution

Equation of the tangent at $(t^{2}, 2 t)$ to the
parabola $y^{2} = 4 x$ is $t y = x + t^{2}$ (1)
and the tangent at $t (- 16 t^{'}, - 8 t^{2})$ to the parabola $x^{2} = - 32 y$
$t^{'} x = y - 8 t^{' 2}$ (2)
Since (1) and (2) represent the same line, comparing we get
$\frac{1}{- t^{'}} = \frac{- t}{1} = \frac{t^{2}}{- 8 t^{' 2}} \Rightarrow t = 2$
Hence the required slope of the tangent is $\frac{1}{t} = \frac{1}{2}$ .

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