If the line a^2 x + ay +1=0, for some real number a, is normal to the curve x...

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Question Id : 11665 | Context :NIMCET 2024

If the line

$a^2 x + ay +1=0$ , for some real number

$a$ , is normal to the curve

$xy=1$ then

🎥 Video solution / Text Solution of this question is given below:

The line $a^2x + ay + 1 = 0$ is normal to the curve $xy = 1$ . Find possible values of $a \in \mathbb{R}$ .

Rewrite: $y = -a x - \frac{1}{a}$ → slope = $-a$

$xy = 1 \Rightarrow \frac{dy}{dx} = -\frac{y}{x}$ Slope of normal = $\frac{x}{y}$

$-a = \frac{x}{y} \Rightarrow x = -a y$

$xy = 1 \Rightarrow (-a y)(y) = 1 \Rightarrow y^2 = -\frac{1}{a}$

For real $y$ , we need $a < 0$

$\boxed{a < 0}$

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