Question

A circle whose centre is the point of intersection of the lines 2x − 3y + 4 = 0 and 3x + 4y − 5 = 0 passes through the origin. Find its equation.

Answer 1

Let the required equation of the circle be ${\left(x-h\right)}^2+{\left(y-k\right)}^2=a^2$.
The point of intersection of the lines 2x − 3y + 4 = 0 and 3x + 4y − 5 = 0 is $\left(\frac{-1}{17},\frac{22}{17}\right)$.
∴ Centre = $\left(\frac{-1}{17},\frac{22}{17}\right)$

Also, the circle passes through the origin.

∴ $a^2={\left(\frac{1}{17}\right)}^2+{\left(\frac{22}{17}\right)}^2=\frac{485}{289}$

Hence, the required equation of the circle is ${\left(x+\frac{1}{17}\right)}^2+{\left(y-\frac{22}{17}\right)}^2=\frac{485}{289}$.