Question

If $A$ and $B$ are two fixed points, then the locus of a point which moves in such a way that the $\angle APB$ is a right angle, is

• A. A circle
• B. An ellipse
• C. A parabola
• D. None of these

Answer 1

The correct option is A A circle

Let $A(x_1,y_1)$ and $B(x_2,y_2)$ be two fixed points and $P(h,k)$ be a variable point such that
$\angle APB=\frac{\pi }{2}$
Then, slope of $AP\times$ slope of $BP=-1$
$\Rightarrow \frac{k-y_1}{h-x_1}\times \frac{k-y_2}{h-x_2}=-1$
$\Rightarrow (h-x_1)(h-x_2)+(k-y_1)(k-y_2)=0$
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.