Question

Find the ratio of the length of the tangents from any point on the circle $15x^2+15y^2-48x+64y=0$ to the two circles $5x^2+5y^2-24x+32y+75=0,$ $5x^2+5y^2-48x+64y+300=0.$

• A. $1:2$
• B. $2:5$
• C. $1:3$
• D. $1:4$

Answer 1

The correct option is A $1:2$

Let $P(h,k)$ be a point on the circle $15x^2+15y^2-48x+64y=0.$
$\therefore h^2+k^2-\frac{48}{15}h+\frac{64}{15}k=0\Rightarrow 3(h^2+k^2)=(\frac{48}{5}h-\frac{64}{5}k)\cdots \left(1\right)$
Now let the tangent lengths be $P{T}_1$ and $P{T}_2$ from $P(h,k)$ to $5x^2+5y^2-24x+32y+75=0$ and $5x^2+5y^2-48x+64y+300=0,$ respectively. Then
$P{T}_1=\sqrt{h^2+k^2-\frac{24}{5}h+\frac{32}{5}k+15}=\sqrt{15-\frac{1}{2}(h^2+k^2)}\left[\text{From}\right(1\left)\right]$
and
$P{T}_2=\sqrt{h^2+k^2-\frac{48}{5}h+\frac{64}{5}k+60}=\sqrt{60-2(h^2+k^2)}\left[\text{From}\right(1\left)\right]=2\sqrt{15-\frac{1}{2}(h^2+k^2)}$

$\therefore P{T}_1:P{T}_2=1:2$