Question

If the line $3x-4y=$$\lambda$ cuts the circle $(x^2+y^2-4x-8y-5=0)$ in two distinct points,

then the limits of $\lambda$ are:

• A. $\left[-35,15\right]$
• B. $\left(-35,15\right)$
• C. $\left[-15,35\right]$
• D. $\left(-15,35\right)$

Answer 1

The correct option is B $\left(-35,15\right)$

perpendicular distance, $d$ between centre of circle and line should be $0\le d<r$ centre of circle

$=(-g,-f)=(\frac{4}{2},\frac{8}{2})$

$=(2,4)$ radius,

$r=\sqrt{f^2+g^2-c}=\sqrt{4+16+5}=5$

$\Rightarrow |\frac{3\left(2\right)-4\left(4\right)-\lambda }{5}\Vert \le r$

$\Rightarrow |10+\lambda |<5r\Rightarrow |\lambda +10|<25$

$\Rightarrow -25<\lambda +10<25\Rightarrow -35<\lambda <15$