Question

If the circle $x^2+y^2=4x+8y+5$ intersects the line $3x-4y=m$ at two distinct points, then the number of possible integral values of $m$ is equal to

• A. 51
• B. 50
• C. 49
• D. 48

Answer 1

Answer: (C)

The equation of the circle is $x^2+y^2-4x-8y-5=0$
Its centre is $\left(2,4\right)$ and radius is $\sqrt{4+16+5}=5$ units
It the circle intersects the line $3x-4y=m$ at two distinct points, then the length of the perpendicular from the centre is less than the radius
i.e., $\frac{|6-16-m|}{5}<5$
$\Rightarrow \left|10+m\right|<25\Rightarrow -25<m+10<25\Rightarrow -35<m<15$
Hence, the total number of integral values of $m$ is $49$