Question

The circle $x^2+y^2=4x+8y+5$ intersects the line $3x-4y=m$ at two distinct points if :-

• A. $-85<m<-35$
• B. $-35<m<15$
• C. $15<m<65$
• D. $35<m<85$

Answer 1

Answer: (B)

$-35<m<15$

circle $x^2+y^2-4x-8y-5=0$

standard circle : $a{x}^2+2nxy+b{y}^2+2gx+2fy+c=0$

on comparing -

center $(2,4)$

Radius$=\sqrt{4+16+5}=\sqrt{25}=5$

If the circle is interacting line $3x-4y=m$ at distinct

points:

$\Rightarrow$ Length of perpendicular from center to

line $<$ radius

$\Rightarrow \left|\frac{6-16-m}{5}\right|<25$

$\Rightarrow |10+m|<25$

$\Rightarrow -25<m+10<25$

$\Rightarrow -35<m<15$