The number of integral values of k for which
Question:

The number of integral values of $k$ for which the line, $3 x+4 y=k$ intersects the circle, $x^{2}+y^{2}-2 x-4 y+4=0$ at two distinct points is_________.

Solution:

The given circle is $x^{2}+y^{2}-2 x-4 y+4=0$

$\therefore$ Centre of circle $(1,2), r=1$.

If line cuts circle then $p<r$, where $p=\left|\frac{a x_{1}+b y_{1}+c}{\sqrt{a^{2}+b^{2}}}\right|$

$\Rightarrow\left|\frac{3+8-k}{5}\right|<1 \Rightarrow k \in(6,16)$

$k=7,8,9,10,11,12,13,14,15$