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The locus of the mid-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

Option: 1

\frac{r}{2}


Option: 2

\frac{2 r}{3}


Option: 3

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


Option: 4

\frac{r}{\sqrt{3}}


Answers (1)

best_answer

Let the equation of the circle be x^2+y^2=r^2. The chord which subtends an angle \frac{\pi}{4} at the circumference will subtend a right angle at the centre.

So, chord joining A(r, 0) and B(0, r) subtends a right angle at the centre (0,0). Mid point of AB is \left(\frac{r}{2}, \frac{r}{2}\right).

\therefore O C=\frac{r}{\sqrt{2}} , which is radius of locus of C.

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