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Q.
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
NTA AbhyasNTA Abhyas 2022
Solution:
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., $\frac{\left|6 - 16 - m\right|}{5} < 5$
$\Rightarrow \left|10 + m\right| < 25\Rightarrow -25 < m+10 < 25\Rightarrow -35 < m < 15$
Hence, the total number of integral values of $m$ is $49$