The equation of a circle is (x−6)² + (y+a)² = 18. The line with equation y = 2a−x is a tangent to the circle.

Mark scheme

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

1. [M1*] (x-6)² + (2a-x+a)² = 18 Replacing y with 2a – x in the circle equation, condone incorrect expansion before substitution.
2. [A1] 2x² -12x−6ax+9a² +36-18[=0] All terms collected on one side of the equation. May be implied by the discriminant.
3. [DM1] (12+6a)²-4×2×(9a² +18) [= 0] Correct use of "b² - 4ac" from their 3 term quadratic equation in x, with an x term of the form (m+na)x with both m and n ≠ 0.
4. [A1] -36a² +144a[+0=0]
5. [A1] a = 0, a = 4
6. [B1] [Centre is] (6, -4) or [Point of intersection is] (9, -1)
7. [B1] [Gradient of diameter] = 1
8. [B1FT] y+4=x-6 or y+1=x-9 [leading to y = x -10] FT on their point of intersection or their centre with an x co-ordinate of ±6 and gradient = 1.