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, $x^2+2x\sin xy+1=0$
$\Rightarrow x^2+2x\sin xy+si{n}^{2}xy+1-{\sin }^{2}xy=0$
$\Rightarrow (x+\sin xy{)}^2+{\cos }^{2}xy=0$
$\therefore x+\sin xy=0$ 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