2024 TMJC Promo Q3

The curve \(C_1\) has equation \(y = 2x + 3 + \frac{12}{x - 2}\), \(x \neq 2\).

1. Sketch \(C_1\), showing clearly the equations of asymptotes, the coordinates of the axial intercepts and the turning points, if any.[4]
2. By completing the square for both \(x\) and \(y\), give a geometrical interpretation of the curve \(C_2\) with equation \(x^2 - 4x + y^2 + 6y - 3 = 0\).
   Hence write down the number of real roots for the equation \[(x - 2)^2 + \left(2x + 6 + \frac{12}{x - 2}\right)^2 = 4^2.\][3]