The circle x2+y2=4x+8y+5 intersects the line 3x−4y=m at two distinct points if:
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Official solution
Circle x2+y2−4x−8y−5=0
Center =(2,4), Radius \[ = \sqrt {4 + 16 + 5} = 5\]
If circle is intersecting line 3x−4y=m, at two distinct points.
\[ \Rightarrow \] length of perpendicular from center to the line < radius
\[ \Rightarrow {{\left| {6 - 16 - m} \right|} \over 5} < 5 \Rightarrow \left| {10 + m} \right| < 25\]
\[ \Rightarrow - 25 < m + 10 < 25 \Rightarrow - 35 < m < 15\]