Question

Find the range of $\lambda$ if the line $3x-4y=\lambda$ meets the circle $x^2+y^2-4x-8y-5=0$ in real points.

Answer 1

Eliminating y,
$25x^2-2(3\lambda +80)x+({\lambda }^2+32\lambda -80)=0$
for real points
$B^2-4AC>0$ $\therefore {\lambda }^2+20\lambda -525<0$
or $(\lambda +35)(\lambda -15)<0\Rightarrow -35<\lambda <15.$