A) a circle
B) an ellipse
C) a parabola
D) None of these
Correct Answer: A
Solution :
Let \[A({{x}_{1}},{{y}_{1}}),\,B({{x}_{2}},{{y}_{2}})\] be two fixed points and let \[P(h,k)\] be a variable point such that \[\angle APB=\frac{\pi }{2}.\] Then, slope of\[AP\times \]slope of \[BP=1\] \[\Rightarrow \] \[\frac{k-{{y}_{1}}}{h-{{x}_{1}}}.\,\frac{k-{{y}_{2}}}{h-{{x}_{2}}}=-1\] \[\Rightarrow \] \[(h-{{x}_{1}})\,(h-{{x}_{2}})+(k-{{y}_{1}})(k-{{y}_{2}})=0\] Hence, locus of \[(h,k)\] is \[(x-{{x}_{1}})(x-{{x}_{2}})+(y-{{y}_{1}})(y-{{y}_{2}})=0\] Which is a circle having AB as diameter.
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