All points (x, y) in plane satisfying equation x^2+2 x sin (x y)+1=0 lie on

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4-2.Quadratic Equations and Inequations

hard

All the points $(x, y)$ in the plane satisfying the equation $x^2+2 x \sin (x y)+1=0$ lie on

A

a pair of straight lines

B

a family of hyperbolas

C

a parabola

D

an cllipse

(KVPY-2011)

Solution

(a)

We have, $x^2+2 x \sin x y+1=0$

$\Rightarrow x^2+2 x \sin x y+\sin ^2 x y+1-\sin ^2 x y=0$

$\Rightarrow \quad(x+\sin x y)^2+\cos ^2 x y=0$

$\therefore x+\sin x y=0$ and $\cos ^2 x y=0$

$\begin{aligned} \cos ^2 x y &=0 \\ \Rightarrow \quad x y &=(2 n+1) \frac{\pi}{2} \end{aligned}$

$\Rightarrow \quad x+1=0$

$\left[\because \cos ^2 x y=0 \Rightarrow \sin x y=1\right]$

$\Rightarrow \quad x=-1$

$\therefore \quad y=-(2 n+1) \frac{\pi}{2}$

Standard 11

Mathematics

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