A circle is drawn touching the \(X\)-axis, with its centre at the point of reflection of \((m, n)\) on the…
A circle is drawn touching the \(X\)-axis, with its centre at the point of reflection of \((m, n)\) on the line \(y-x=0\). Then the equation of the circle is
\(x^2+y^2-2 m x-2 n y+m^2=0\)
\(x^2+y^2-2 m x+2 n y+m^2=0\)
\(x^2+y^2+2 n x-2 m y-n^2=0\)
\(x^2+y^2-2 n x-2 m y+n^2=0\)
Solution
The point of reflection of \((m, n)\) on the line \(y-x=0\) is \((n, m)\), so equation of circle having centre \((n, m)\) and let radius \(r\) is
\((x-n)^2+(y-m)^2=r^2\)...(i)
\(\because\) Circle (i) touches the \(X\)-axis, so \(r=m\)
So, equation of required circle is
\(\begin{aligned}
(x-n)^2+(y-m)^2 & =m^2 \\
\text {or } x^2+y^2-2 n x-2 m y+n^2 & =0
\end{aligned}\)