Question

The locus of the middle points of those chords of the circle $x ^ { 2 } + y ^ { 2 } = 4$ which subtend a right angle at the origin is

• A. $x ^ { 2 } + y ^ { 2 } - 2 x - 2 y = 0$
• B. $x ^ { 2 } + y ^ { 2 } = 4$
• C. $x ^ { 2 } + y ^ { 2 } = 2$
• D. $( x - 1 ) ^ { 2 } + ( y - 2 ) ^ { 2 } = 5$

Answer 1

Answer: (C)

$x ^ { 2 } + y ^ { 2 } = 2$

Answer 2

Let the mid-point of chord is (h, k). Also radius of circle is 2. Therefore

$\frac { O C } { O B } = \cos 45 ^ { \circ } \Rightarrow \frac { \sqrt { h ^ { 2 } + k ^ { 2 } } } { 2 } = \frac { 1 } { \sqrt { 2 } } \Rightarrow h ^ { 2 } + k ^ { 2 } = 2$

Hence locus is $x ^ { 2 } + y ^ { 2 } = 2$