The equation of the common tangent to the curves y 2 = 8 x and x y = - 1 is [2002]

Question

The equation of the common tangent to the curves $y^{2} = 8 x$ and $x y = - 1$ is [2002]

Smart Achievers · Accepted Answer

(4)

$Given equation of curves are$

$y^{2} = 8 x ...(i)$

and $x y = - 1 ...(ii)$

If $m$ is the slope of tangent to (i), then equation of tangent is $y = m x + \frac{2}{m}$

If this tangent is also a tangent to (ii), then putting value of $y$ in curve (ii)

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

We should get repeated roots for the equation (condition of tangency)

$\Rightarrow D = 0$ $∴$ ${(2)}^{2} - 4 m^{2} \cdot m = 0$ $\Rightarrow m^{3} = 1$ $\Rightarrow m = 1$

Hence equation of required tangent is $y = x + 2$

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