The line y = 3x-2 intersects the curve 2x²-xy+y² = 2 at the points A and B. The point C with coordinates (k, 7/8) lies on the perpendicular bisector of the line AB. Find the exact value of k.

Mark scheme

Marking points from the official mark scheme (May/June 2024). The mark or mark type is shown in brackets.

1. [M1] 2x²-x(3x-2)+(3x-2)² = 2 or 2( (y+2)/3 )² - y( (y+2)/3 ) + y² = 2 For a quadratic equation in one variable
2. [A1] 4x²-5x+1=0 oe or 4y² + y-5=0 oe Allow multiples but must be a 3 term equation = 0
3. [A1] x = 1/4, x = 1
4. [A1] y = -5/4, y = 1
5. [M1] Midpoint: (5/8, 1/8) Allow for use of their coordinates
6. [M1] Gradient of AB = 3 Allow for use of their coordinates
7. [M1] Perpendicular gradient: -1/3 Dep on previous M mark
8. [M1] Perpendicular equation: y+5/8 = -1/3 (x-1/8) Allow for their perp gradient and their midpoint
9. [A1] k = 19/8 or -2.375 oe
10. [B1] x-coordinate: 29/8 or 3.625 oe
11. [B1] y-coordinate: 9/8 or 1.125 oe