Question

A line (with constant term $k$) perpendicular to $4x+3y+2=0$, is tangent to circle with integral radius (radius as integer). If the ratio of minimum to maximum possible distance from origin to such tangents is $1:3$, then possible value of $k$ such that the radius of circle is minimum

• A. $3$
• B. $-3$
• C. $5$
• D. $-5$

Answer 1

Answer: (C), (D)

$-5$

$3x-4y+k=0$ is a tangent


ratio of minimum to maximum distance from origin is:
$\frac{k_1}{k_2}=\frac{1}{3}$
diameter $=\frac{|k_2-k_1|}{5}=\frac{|2k_1|}{5}$
radius $=\frac{|k_1|}{5}$
radius minimum $\Rightarrow k=\pm 5$