Theory question 109
Polynomials(a) The roots of the equation x2 + mx + 11 = 0 are α and β, where m is a constant. If α2 + β2 = 27, find the value of m.
(b) The line 2x + 3y = 1 intersects the circle 2x2 + 2y2 + 4x + 9y − 9 = 0 at points P and Q where Q lies in the fourth quadrant. Find the coordinates of P and Q.
Show the model answer
Model answer
(a) α + β = −m, αβ = 11
α² + β² = (α + β)² − 2αβ = m² − 22 = 27
m² = 49, m = ±7
(b) y = (1 − 2x)/3; substitute and multiply by 9:
18x² + 2(1 − 2x)² + 36x + 27(1 − 2x) − 81 = 0
26x² − 26x − 52 = 0, x² − x − 2 = 0, x = 2 or x = −1
x = 2: y = −1 (fourth quadrant), so Q(2, −1)
x = −1: y = 1, so P(−1, 1)
Explanation
Use sum and product of roots in (a); in (b) substitute the line into the circle.