AB is chord of the circle x² +y² = 25 whose middle point is M. Then match the following loci of M in various condition
Column-I Column-II
A. when AB subtends 60° at centre P. $x^{2} + y^{2} + 2 y = 0$
B. when AB always passes through (0, - 2) Q. $x^{2} + y^{2} = 9$
C. when AB has constant length 8 R. $x^{2} + y^{2} = \frac{75}{4}$

Moderate

Solution

Let M(x₁,y₁) be a mid point of AB
$A) \sin 60 ° = \frac{\sqrt{x_{1}^{2} + y_{1}^{2}}}{5} \Rightarrow \frac{\sqrt{3}}{2} = \frac{\sqrt{(x^{2} + y^{2})}}{5} \Rightarrow x^{2} + y^{2} = \frac{75}{4}$
$B) (\frac{y_{1} + 2}{x_{1}}) (\frac{y_{1}}{x_{1}}) = - 1 \Rightarrow x_{1}^{2} + y_{1}^{2} + 2 y_{1} = 0$
$C) 2 \sqrt{25 - d^{2}} = 8 \Rightarrow x^{2} + y^{2} = 9$

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SCAN ME

Column-I	Column-II
A. when AB subtends 60° at centre	P. $x^{2} + y^{2} + 2 y = 0$
B. when AB always passes through (0, - 2)	Q. $x^{2} + y^{2} = 9$
C. when AB has constant length 8	R. $x^{2} + y^{2} = \frac{75}{4}$