The line y = kx – 4, where k is a positive constant, passes through the point P (0, -4) and is a tangent to the curve x² + y² - 2y = 8 at the point T. Find

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2016). The mark or mark type is shown in brackets.

1. [M1] (i) tangent touches circle x²+(kx-4)²-2(kx-4) = 8 eliminate y or x allow unsimplified
2. [A1] k²x²+x²-8kx−2kx+16=0 or better
3. [DM1] Equal roots as tangent touches circle : b² = 4ac use of discriminant on 3 term quadratic soi
4. (-10k)² = 4(k²+1)×16
5. [A1] 36k² = 64
6. [A1] k = +4/3 only oe any inequality loses last A1
7. [M1] (ii) x = -b/(2a) → x = 10k/(2(k²+1)) use x = -b/(2a)
8. x = 12/5, y = -4/5 [A1A1]
9. OR tangent y = 4/3 x-4 cuts radius [M1] find equation of radius and attempt to solve with tangent
10. y = -3/4 x+1
11. at x = 12/5 [A1]
12. y = -4/5 [A1]
13. OR Obtain 25x² - 120x + 144 = 0 oe [M1] obtain any 3 term quadratic using their non zero k and reach x = ...
14. (5x-12)(5x-12) = 0
15. x = 12/5 → y = -4/5 [A1A1]
16. [M1A1] (iii) TP = √(0-2.4)²+(-4+0.8)² = 4 M1 for using their T and (0,-4). Signs must be correct.