Question

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

• A. a pair of straight lines
• B. a family of hyperbolas
• C. a parabola
• D. an ellipse

Answer 1

Answer: (A)

We have, $x^{2}+2x\,\sin \,xy +1=0$
$\Rightarrow x^{2}+2x\,\sin \,xy +sin^{2}\,xy +1-\sin^{2}\,xy=0$
$\Rightarrow (x+\sin \,xy)^{2} + \cos^{2}\,xy =0$
$\therefore x+\sin\,xy =0$ and $\cos^{2}\,xy=0$
$\cos^{2}\,xy =0$
$\Rightarrow xy =(2n+1) \frac{\pi}{2}$
$\Rightarrow x+1=0$
$[\because \cos^{2}\,xy =0 \Rightarrow \,\sin\,xy=1]$
$\Rightarrow x=-1$
$\therefore y=-(2n+1) \frac{\pi}{2}$
which represent the pair of straight line