Question

The locus of a point, which is such that the length of the tangents from its two concentric circles of radii $a$ and $b$ are inversely proportional to their radii, is a circle $C$ of area $16\pi$

• A. centre of $C$ is at the centre of the given circles
• B. centre of $C$ is at a distance $4$ from the centre of the given circles
• C. $a^2+b^2=16$
• D. $a^2+b^2=4$

Answer 1

Answer: (A), (C)

centre of $C$ is at the centre of the given circles
$a^2+b^2=16$

Let $a>b$
Let the first circles be $x^2+y^2=a^2$ and $x^2+y^2=b^2$.
Let $t_1$ and $t_2$ be the length of the tangents drawn from $P(h,k)$.
As per the given conditions,
$t_1\times a=t_2\times b$ -- (i)
By Pythagoras theorem,
$t_1^2=h^2+k^2-a^2$
$t_2^2=h^2+k^2-b^2$
Squaring (i) ,
$t_1^2{a}^2=t_2^2{b}^2$
$(h^2+k^2-a^2)a^2=(h^2+k^2-b^2)b^2$
$(h^2+k^2)(a^2-b^2)=a^4-b^4$
$h^2+k^2=a^2+b^2$
Since the area is 16$\pi$, the radius is equal to $4$
Hence,
$a^2+b^2=16$
Also, C is the centre of the circle
Hence, options A and C are correct.