First slide

Parabola

Question

If m is the slope of a common tangent of the parabola $y^{2} = 16 x$ and the circle $x^{2} + y^{2} = 8$ , then $m^{2}$ is equal to

Moderate

A

1
B

2
C

4
D

8

Solution

Equation of a tangent to the parabola $y^{2} = 16 x$ is $y = m x + \frac{4}{m}$
If this touches the circle of $x^{2} + y^{2} = 8$ , then the distance of the centre $(0, 0)$ of the circle from this tangent is equal to the radius $2 \sqrt{2}$ of the circle.
$\begin{array}{ll} \frac{4}{m} \\ \Rightarrow \sqrt{1 + m^{2}} = 2 \sqrt{2} \\ \Rightarrow 2 = m^{2} + m^{4} \\ \Rightarrow (m^{2} + 2) (m^{2} - 1) = 0 \Rightarrow m^{2} = 1 . \end{array}$

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