2025 RI Promo Q7

The curve \(C_1\) has equation \(y=\frac{ax^2+bx+3}{x+2}\), where \(a\) and \(b\) are constants. It is given that \(C_1\) has an asymptote \(y=x-3\).

1. State the value of \(a\) and show that \(b=-1\).[3]

2. Using an algebraic method, find the set of values that \(y\) cannot take.

   [3]

The locus of a point is defined as the path traced out by that point as it moves.

3. Let \(P(x, y_1)\) be a point on the curve \(C_1\) and \(Q(x, y_2)\) be a point on the curve \(C_2\) with equation \(y=\frac{3x-x^2}{x+2}\). Find the cartesian equation of the locus of the midpoint of \(P\) and \(Q\) as \(x\) varies.[2]