Find the minimum radius of the circle which is orthogonal to both the circles \(x^2+y^2+4 x+3=0\) and…
Find the minimum radius of the circle which is orthogonal to both the circles \(x^2+y^2+4 x+3=0\) and \(x^2+y^2-12 x+35=0\).
1
4
\(\sqrt{17}\)
\(\sqrt{15}\)
Solution
Centre of circle cuts the circles
\(\begin{aligned}
S_1: x^2+y^2+4 x+3 & =0 \\
S_2: x^2+y^2-12 x+35 & =0
\end{aligned}\)
Radical Axis is \(S_1-S_2=0 \Rightarrow x=2\)
and \((2,0)\) lies on the line segment joining centres of circles \(S_1\) and \(S_2\)
Minimum radius \(=\) length of tangent from \((2,0)\)
\(\text { to circle } \begin{aligned}
S_1 & =0 \text { (or) } S_2=0 \\
& =\sqrt{4+8+3}=\sqrt{15}
\end{aligned}\)
Hence, option (d) is correct.