Find the equation of the normal to the curve \(5x{y^2} - 2{x^2} = 18\) at the point (1, 2) .

\(5{y^2} + 10xy\frac{{{\text{d}}y}}{{{\text{d}}x}} - 4x = 0\)     A1A1A1

Note: Award A1A1 for correct differentiation of \(5x{y^2}\).

A1 for correct differentiation of \( - 2{x^2}\) and 18.

At the point (1, 2), \(20 + 20\frac{{{\text{d}}y}}{{{\text{d}}x}} - 4 = 0\)

\( \Rightarrow \frac{{{\text{d}}y}}{{{\text{d}}x}} = - \frac{4}{5}\)     (A1)

Gradient of normal \( = \frac{5}{4}\)     A1

Equation of normal \(y - 2 = \frac{5}{4}(x - 1)\)     M1

\(y = \frac{5}{4}x - \frac{5}{4} + \frac{8}{4}\)

\(y = \frac{5}{4}x + \frac{3}{4}\,\,\,\,\,(4y = 5x + 3)\)     A1

[7 marks]

It was pleasing to see that a significant number of candidates understood that implicit differentiation was required and that they were able to make a reasonable attempt at this. A small number of candidates tried to make the equation explicit. This method will work, but most candidates who attempted this made either arithmetic or algebraic errors, which stopped them from gaining the correct answer.