Question

Find the values of $p$ for which the power of a point $(2,5)$ is negative with respect to a circle $x^2+y^2-8x-12y+p=0$ which neither touches the axes nor cuts them.

Answer 1

Given that the circle, having centre at $(4,6)$, neither touches the axes nor cuts them

$\therefore$ radius $<4$

$\Rightarrow \sqrt{16+36-p}<4\Rightarrow p>36\dots \left(1\right)$

And power of point $(2,5)$ is negative

$\therefore 4+25-16-60+p<0\Rightarrow p<47\dots \left(2\right)$

From $\left(1\right)$ and (2), we get

$\Rightarrow p\in (36,47)$