O-LevelAdditional MathematicsEquations, inequalities and graphs4037/21 · May/June 2023 · Q119 Marks
The line with equation x+3y = k, where k is a positive constant, is a tangent to the curve with equation x²+y²+2y-9 = 0. Find the value of k and hence find the coordinates of the point where the line touches the curve.
- The line with equation x+3y = k, where k is a positive constant, is a tangent to the curve with equation x²+y²+2y-9 = 0. Find the value of k and hence find the coordinates of the point where the line touches the curve.[9]
Mark scheme
Marking points from the official mark scheme (May/June 2023). The mark or mark type is shown in brackets.
- [M1] Eliminates one variable (k-3y)² + y² + 2y - 9 = 0
- [A1] 10y² + (2–6k)y + (k² – 9) = 0 soi
- [M1] Uses b²-4ac = 0 with their 3-term quadratic: (2-6k)² - 4(10)(k²-9) = 0
- [A1] -4k² - 24k +364 = 0
- [M1] Factorises -4k² – 24k +364 or solves their -4k²-24k +364 = 0
- [A1] k = 7 only
- [M1] Uses their k in 10y² + (2–6k)y + (k² – 9) = 0 oe
- [A1] y = 2 only
- [A1] x = 1 only
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