Question

If the pair of lines given by $(x^2+y^2){\cos }^2\phi =(x\cos \phi +y\sin \phi {)}^2$ are perpendicular to each other, then $\phi =$

• A. $0$
• B. $\frac{\pi }{4}$
• C. $\frac{\pi }{3}$
• D. $3\frac{\pi }{4}$

Answer 1

The correct option is B $\frac{\pi }{4}$

$x^2{\cos }^2\phi +y^2{\cos }^2\phi =x^2{\cos }^2\phi +y^2{\sin }^2\phi +2xy\cos \phi \sin \phi$

$\Rightarrow y^2({\cos }^2\phi -{\sin }^2\phi )-2xy\cos \phi \sin \phi =0$

$\Rightarrow y^2\cos 2\phi -xy\sin 2\phi =0$

$\Rightarrow y(y\cos 2\phi -x\sin 2\phi )=0$

$\therefore y=0$ and $y=x\tan 2\phi$

As these two lines are perpendicular

$\tan 2\phi =\tan \frac{\pi }{2}$

$\Rightarrow \phi =\frac{\pi }{4}$