Theory question 111
Differentiation(a) Find the equation of the normal to the curve y = (x2 − x + 1)(x − 2) at the point where the curve cuts the x-axis. (b) The coordinates of the points P, Q and R are (−1, 2), (5, 1) and (3, −4) respectively. Find the equation of the line joining Q and the midpoint of PR.
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Model answer
(a) x² − x + 1 has no real roots, so the curve cuts the x-axis only at (2, 0).
dy/dx = (2x − 1)(x − 2) + (x² − x + 1) = 3 at x = 2; normal gradient −1/3.
y = −(x − 2)/3, i.e. x + 3y − 2 = 0.
(b) Midpoint of PR = (1, −1). Gradient to Q(5, 1) = 2/4 = ½.
y + 1 = ½(x − 1), i.e. x − 2y − 3 = 0.
Explanation
Use the product rule for the gradient.