The equation of a curve is 2x²y – xy² = a³, where a is a positive constant. Show that there is only one point on the curve at which the tangent is parallel to the x-axis and find the y-coordinate of this point.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2019). The mark or mark type is shown in brackets.

1. [B1] State 4xy + 2x² dy/dx, or equivalent, as derivative of 2x²y
2. [B1] State y² + 2xy dy/dx, or equivalent, as derivative of xy²
3. [*M1] Equate attempted derivative of LHS to zero and set dy/dx equal to zero (or set numerator equal to zero)
4. [B1] Reject y = 0
5. [A1] Obtain y = 4x
6. [DM1] Obtain an equation in y (or in x) and solve for y (or for x) in terms of a
7. [A1] Obtain y = -2a
8. [M1] Rewrite as y = a³/(2x² - xy) and differentiate
9. [A1] Obtain correct derivative (in any form)
10. [B1] x=0 and 2x = y both give a = 0