Question

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

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

Answer 1

Answer: (A)

Circle $x^2 + y^2 - 4x - 8y - 5 = 0$
Centre $= (2, 4)$, Radius $= \sqrt{4+16+5} = 5$
If circle is intersecting line $3x - 4y = m$
at two distinct points.
$\Rightarrow $ length of perpendicular from centre < radius
$\Rightarrow \frac{\left|6-16-m\right|}{5} < 5$
$\Rightarrow \left|10+m\right| < 25$
$\Rightarrow -25 < m + 10 < 25$
$\Rightarrow -35 < m < 15.$