Question

If the points $(\lambda ,-\lambda )$ lies inside the circle $x^2+y^2-4x+2y-8=0$, then find the range of $\lambda$.

Answer 1

If the given points lie inside the Circle then

$f(x,y{)}_{at(\lambda ,-\lambda )}<0\Rightarrow {\lambda }^2+{\lambda }^2-4\lambda -2\lambda -8<0\Rightarrow 2{\lambda }^2-6\lambda -8<0\Rightarrow 2(\lambda +1\left)\right(\lambda -4)<0\Rightarrow \lambda ϵ(-1,4)$