Question

STATEMENT - 1 : Locus of mid point of chords of circle $x^2+y^2=4$ which subtends angle of $\frac{\pi }{2}$ at origin is $x^2+y^2=1.$
STATEMENT - 2 : If any chord of circle $x^2+y^2=r^2$ subtends an angle ${}^{\prime }{\theta }^{\prime }$ at center, then its mid point always lies on $x^2+y^2=r^2{\cos }^2\left(\frac{\theta }{2}\right)$

• A. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is a correct explanation for STATEMENT-1
• B. STATEMENT-1 is True, STATEMENT-2 is True; STATEMENT-2 is NOT a correct explanation for STATEMENT-1
• C. STATEMENT-1 is True, STATEMENT-2 is False
• D. STATEMENT-1 is False, STATEMENT-2 is True

Answer 1

Answer: (D)

STATEMENT-1 is False, STATEMENT-2 is True

$\therefore$ locus of point $(h,k)$ is $x^2+y^2=r^2{\cos }^2\left(\frac{\theta }{2}\right)$
$\therefore$ Statement-1 is false and Statement-2 is true.
$\frac{\sqrt{h^2+k^2}}{r}=\cos \left(\frac{\theta }{2}\right)$
$h^2+k^2=r^2{\cos }^2\left(\frac{\theta }{2}\right)$