First slide

Parabola

Question

$Equation of common tangent to y^{2} = 8 x and x^{2} + y^{2} = 2 is$

Moderate

A

$y = \pm (x + 1)$
B

$y = \pm (x + 2)$
C

$y = \pm (x - 1)$
D

$y = \pm (x - 2)$

Solution

$Equation of tangent to y^{2} = 8 x is y = mx + \frac{2}{m} \Rightarrow m^{2} x - my + 2 = 0 is also tangent to x^{2} + y^{2} = 2$
$\begin{array}{l} \Rightarrow \frac{| 2 |}{\sqrt{m^{4} + m^{2}}} = \sqrt{2} \Rightarrow m^{4} + m^{2} - 2 = 0 \\ \Rightarrow (m^{2} + 2) (m^{2} - 1) = 0 \Rightarrow m = \pm 1 \\ ∴ required equation y = (\pm 1) x + \frac{2}{(\pm 1)} \Rightarrow y = \pm (x + 2) \end{array}$

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