Question

The point $(1,4)$ lies inside the circle $x^2+y^2-6x-10y+\lambda$, If the circle neither touched nor cuts the axes, then the difference between the maximum and minimum possible values of $\lambda$ is

Answer 1

eqn of circle : $x^2+y^2-6x-10y+\lambda$

radius of circle $\Rightarrow (-g,-f)\Rightarrow (3,5)$

$\therefore$ radius of value $73$

$\therefore {75}^2,\lambda 725$

$\to$ Also, $(1,4)$ lies inside circle, So $S_1<0$

$\therefore 1^2+4^2-6\times 1-10\left(4\right)+\lambda <0$

$1+16-6-40+\lambda <0$

$\lambda <29$

$\therefore (25<\lambda <29)$