Let A be a point on the x-axis. Common tangents are drawn from A to the curves x 2 + y 2 = 8 and y 2 = 16 x . If one of these tangents touches the two curves at Q and R, then ( Q R ) 2 is equal to [2023]

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Solution

Q.
Let A be a point on the x-axis. Common tangents are drawn from A to the curves $x^{2} + y^{2} = 8$ and $y^{2} = 16 x$ . If one of these tangents touches the two curves at Q and R, then ${(Q R)}^{2}$ is equal to [2023]

1 76

2 81

3 72

4 64

Ans.
(3)

Given curves are $x^{2} + y^{2} = 8$ and $y^{2} = 16 x$

Equation of tangent to the parabola in slope form is:

$y = m x + \frac{a}{x} \Rightarrow y = m x + \frac{4}{m}$

Length of perpendicular from (0, 0) to the point of tangency is equal to the length of radius of circle.

$\Rightarrow$ $| \frac{\frac{4}{m}}{\sqrt{1 + m^{2}}} |$ $= \sqrt{8} \Rightarrow$ $| \frac{1}{m \sqrt{1 + m^{2}}} |$ $= \frac{1}{\sqrt{2}}$

$m^{2} (1 + m^{2}) = 2 \Rightarrow m^{4} + m^{2} - 2 = 0$

$(m^{2} + 2) (m^{2} - 1) = 0$ $∴$ $m = \pm 1$

Point of contact on parabola = $(\frac{a}{m^{2}}, \frac{2 a}{m}) = (4, \pm 8)$

Point of contact on circle is $(- 2, 2) or (2, - 2)$

Distance between $Q (- 2, 2)$ and $R (4, 8)$ is $Q R \Rightarrow Q R^{2} = 72$

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