Question

The locus of the middle points of the chords of the circle of radius $r$ which subtend an angle $\frac{\pi }{4}$ at any point on the circumference of the circle is a concentric circle with radius equal to

• A. $\frac{r}{2}$
• B. $\frac{2r}{2}$
• C. $\frac{r}{\sqrt{2}}$
• D. $\frac{r}{\sqrt{3}}$

Answer 1

Answer: (C)

$\frac{r}{\sqrt{2}}$

Equation of the circle be $x^2+y^2=r^2$.

The chord which subtends an angle $\frac{\pi }{4}$ at the circumference will subtends a right angle at the center.

Chord joining $(r,0)$ and $(0,r)$subtends a right angle at the center so $(h,k)$ the mid-point of the chord is $(\frac{r}{2},\frac{r}{2})$ and locus of $(h,k)$ is $x^2+y^2=\frac{r^2}{2}$

Hence radius is $\frac{r}{\sqrt{2}}$