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Q.
If the length of the tangent from any point on the circle $(x - 3)^2 + (y + 2)^2 = 5r^2$ to the circle $(x -3)^2 + (y + 2)^2 = r^2$ is $16$ units, then the area between the two circles in sq. units is
Let point $P(x_1 , y_1)$ be any point on the circle, therefore it satisfy the circle
$(x_1 - 3)^2 + ( y_1 + 2)^2 = 5r^2\,\,\,\,\,\dots(i)$
The length of the tangent drawn from point
$P(x_1, y_1)$ to the circle $(x - 3)^2 + (y + 2)^2 = r^2$ is
$\sqrt{\left(x_{1} -3\right)^{2} +\left(y_{1} +2\right)^{2} -r^{2}}$
$ = \sqrt{5r^{2} -r^{2}} $ (From (i))
$\Rightarrow 16 =2r $
$\Rightarrow r =8 $
$\therefore $ The area between two circles
$ =\pi\,5r^{2} -\pi r^{2} $
$=4 \pi r^{2} =4\pi\times8^{2} $
$= 256 \pi$ sq unit
