Additional Mathematics – Coordinate geometry of the circle | e-Consult

Substituting \(y=2x+5\) into the circle equation:

\((x+3)^{2}+(2x+3)^{2}=9\)

\(5x^{2}+18x+9=0\)

Discriminant \(D=18^{2}-4\cdot5\cdot9=144>0\); two real solutions:

\(x=\dfrac{-18\pm12}{10}\) → \(x{1}=-\dfrac{3}{5},\; x{2}=-3\).

Corresponding \(y\) values: \(y=2x+5\) gives \(y{1}= \dfrac{19}{5},\; y{2}=-1\).

Since there are two distinct intersection points, the line is a chord of the circle.