question_answer

If \[A\,(5,-1)\] and \[B\,(-3,7)\] are two points on a circle and if \[P(x,y)\] is any point on the circle such that s\[\angle APB=\frac{\pi }{2},\] then the equation of the circle is

A)  \[{{x}^{2}}+{{y}^{2}}-2x+6y-22=0\]

B)  \[{{x}^{2}}+{{y}^{2}}+2x+6y-11=0\]

C)  \[{{x}^{2}}+{{y}^{2}}-2x+6y+22=0\]

D)  \[{{x}^{2}}+{{y}^{2}}-2x-6y-22=0\]

Correct Answer: D

Solution :

Since, \[\angle APB=\frac{\pi }{2}\] Therefore, A and B are the extreme points of the diameter of the circle.   We know that, equation of circle with \[({{x}_{1}},{{y}_{1}})\]and  \[({{x}_{2}},{{y}_{2}})\] be the end points of diameter is \[(x-{{x}_{1}})\,(x-{{x}_{2}})+(y-{{y}_{1}})\,(y-{{y}_{2}})=0\] Therefore, equation of circle with \[(5,-1)\] and \[(-3,7)\] be the end points of diameter is \[(x-5)\,(x+3)+(y+1)\,(y-7)=0\] \[\Rightarrow \] \[{{x}^{2}}-2x-15+{{y}^{2}}-6y-7=0\] \[\Rightarrow \] \[{{x}^{2}}+{{y}^{2}}-2x-6y-22=0\] Which is the required equation of circle.