The curve C₁ has equation y = x² – 4x + 7. The curve C₂ has equation y² = 4x + k, where k is a constant. The tangent to C₁ at the point where x = 3 is also the tangent to C₂ at the point P. Find the value of k and the coordinates of P.

Mark scheme

Marking points from the official mark scheme (May/June 2019). The mark or mark type is shown in brackets.

1. [B1] For C1: dy/dx = 2x - 4 → m = 2
2. [M1] y – ‘their 4' = ‘their m' (x – 3) or using y = mx + c
3. [A1] y - 4 = 2(x - 3) or y = 2x - 2
4. [*M1] Forms an equation in one variable using tangent & C2
5. [*DM1] Uses 'discriminant = 0'
6. [A1] 144 = 16(4 – k) → k = - 5
7. [DM1] Uses k to form a 3 term quadratic in x
8. [A1] x = 3/2 (or 1.5), y = 1(or – 1).
9. [B1] For C1: dy/dx = 2x - 4 → m = 2
10. [M1] y – ‘their 4' = ‘their m' (x – 3) or using y = mx + c
11. [A1] y − 4 = 2( x − 3) or y = 2x−2
12. [*M1] For C2: dy/dx = A(4x+k)^(-1/2)
13. [*DM1] At P: 'their 2' = A(4x+k)^(-1/2) → (x = (1-k)/4 or 4x+k=1)
14. [DM1] From 4x + k = 1 and y² = 4x + k → y² = 1
15. [A1] y = 1(or – 1) and x = 3/2 (or 1/2)
16. [A1] From 4x+k=1, k = −5 ( or − 1)