2010 CJC P1 Q8

The equation of a closed curve is \({{\left( x+2y \right)}^{2}}+3{{\left( x-y \right)}^{2}}=27\).

1. Show, by differentiation, that the gradient of the curve at the point \(\left( x,y \right)\) may be expressed in the form \(\frac{\mathrm{d}y}{\mathrm{d}x}=\frac{y-4x}{7y-x}\).[3]
2. The normal to the curve at the point \(\left( -2,1 \right)\) cuts the \(x\)-axis at \(P\) and the \(y\)-axis at \(Q\). Denoting the origin as \(O\), find the area of the triangle \(OPQ\).[4]
3. Find the equations of the tangents to the curve that are parallel to the \(y\)-axis.[3]