question_answer

The locus of the centre of a circle which passes through the point (0, 0) and cuts off a length 2b from the line x = c is-

A) \[{{y}^{2}}+2cx={{b}^{2}}+{{c}^{2}}\]

B)        \[{{x}^{2}}+cx={{b}^{2}}+{{c}^{2}}\]

C) \[{{y}^{2}}+2cy={{b}^{2}}+{{c}^{2}}\]  

D) None of these

Correct Answer: A

Solution :

[a] Let the centre of circle \[C\left( {{x}_{1}},{{y}_{1}} \right)\] As it passes through (0, 0) its radius \[OC=\sqrt{x_{1}^{2}+y_{1}^{2}}\] \[CB=\sqrt{x_{1}^{2}+y_{1}^{2}}\] AB = 2b (given) CM = length of \[\bot \]from C on line \[x-c=0\] \[CM=\left| \frac{{{x}_{1}}-C}{\sqrt{1}} \right|\] In \[\Delta BCM\] \[\Rightarrow {{b}^{2}}=C{{B}^{2}}-C{{M}^{2}}\] \[\Rightarrow {{b}^{2}}={{x}_{1}}^{2}+{{y}_{1}}^{2}-{{({{x}_{1}}-c)}^{2}}\] \[\therefore \text{ }locus\text{ }{{y}^{2}}+2cx={{b}^{2}}+{{c}^{2}}\]