O-Level Equations and inequalities: worked solution

Question

For what constant k is the line y = kx + 6 tangent to the curve 2x² − xy = 3?

Where the line meets the curve both equations are true at once, so substitute y = kx + 6 into the curve equation 2x² − xy = 3 to eliminate y: 2x² − x(kx + 6) = 3 Expand the bracket, multiplying x through both terms: 2x² − kx² − 6x = 3 Move the 3 across so everything is on one side, then group the x² terms: 2x² − kx² − 6x − 3 = 0 (2 − k)x² − 6x − 3 = 0 This is a quadratic in x. A TANGENT touches the curve at exactly one point, so this quadratic must have one repeated root, which means the discriminant is zero: b² − 4ac = 0 Here a = 2 − k, b = −6 and c = −3. Substitute carefully, watching the two negative signs in 4ac: b² − 4ac = (−6)² − 4(2 − k)(−3) = 36 + 12(2 − k) = 36 + 24 − 12k = 60 − 12k Set this equal to zero and solve: 60 − 12k = 0 12k = 60 k = 5 (Check the quadratic is genuine: a = 2 − k must not be 0, so k ≠ 2. Since k = 5, this is fine.) So k = 5.