Locus of the point of intersection of the pair of perpendicular tangents to the circles $x^2+y^2=1$ and $x^2+y^2=7$ is the director circle of the circle with radius
$(A)\sqrt2\hspace{1cm}(B)2\hspace{1cm}(C)2\sqrt2\hspace{1cm}(D)4$
Since $x^2+y^2=1$ and $x^2+y^2=7$ are the concentric circles with center $O$(say).Let $P$ be any point on the required locus.Let $PT_1$ be the tangent from $P$ to the circle $x^2+y^2=7$ and let $PT_2$ be the tangent from $P$ to the circle $x^2+y^2=1$.Since $PT_1$ and $PT_2$ are perpendicular to each other.
But i dont know how to find the required locus.Please help me.Thanks.
Best Answer
You know it's a circle with center $O$, so you just need a point to find the radius.
You can take a tangent the the first circle: $y=1$ and a perpendicular tangent to the other circle: $x=\sqrt{7}$ , the intersection $(1,\sqrt{7})$ is on the circle: the radius is $$\sqrt{1^2+\sqrt{7}^2}=\sqrt{8}=2\sqrt{2}$$