Theory question 110
Differentiation(a) Find the derivative of y = x2(1 + x)3/2 with respect to x.
(b) The centre of a circle lies on the line 2y − x = 3. If the circle passes through P(2, 3) and Q(6, 7), find its equation.
Show the model answer
Model answer
(a) Product rule: dy/dx = 2x(1 + x)^(3/2) + (3/2)x²(1 + x)^(1/2) = ½x(1 + x)^(1/2)(4 + 7x).
(b) The centre is on the perpendicular bisector of PQ: midpoint (4, 5), gradient −1, so x + y = 9.
With 2y − x = 3: y = 4, x = 5. Centre (5, 4); r² = (5 − 2)² + (4 − 3)² = 10.
(x − 5)² + (y − 4)² = 10, i.e. x² + y² − 10x − 8y + 31 = 0.
Explanation
The centre is the same distance from P and Q, so it lies on their perpendicular bisector.