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

\[\left( x + \frac{1}{17} \right)^2 + \left( y - \frac{22}{17} \right)^2 = \frac{485}{289}\]