Tangent/normal intersection problems

12 A curve has parametric equations \(x = \frac { 1 } { t } , y = 2 t\). The point \(P\) is \(\left( \frac { 1 } { p } , 2 p \right)\).

1. Show that the equation of the tangent at \(P\) can be written as \(y = - 2 p ^ { 2 } x + 4 p\). The tangent to this curve at \(P\) crosses the \(x\)-axis at the point \(A\) and the normal to this curve at \(P\) crosses the \(x\)-axis at the point \(B\).
2. Show that the ratio \(P A : P B\) is \(1 : 2 p ^ { 2 }\). \section*{END OF QUESTION PAPER}

11 Fig. 11 shows the curve with parametric equations $$x = 2 \cos \theta , y = \sin \theta , 0 \leq \theta \leq 2 \pi .$$ The point P has parameter \(\frac { 1 } { 4 } \pi\). The tangent at P to the curve meets the axes at A and B . \begin{figure}[h] \end{figure}

1. Show that the equation of the line AB is \(x + 2 y = 2 \sqrt { 2 }\).
2. Determine the area of the triangle AOB .

A curve has parametric equations $$x = t(t - 1), \quad y = \frac{4t}{1-t}, \quad t \neq 1.$$

1. Find \(\frac{dy}{dx}\) in terms of \(t\). [4]

The point \(P\) on the curve has parameter \(t = -1\).

1. Show that the tangent to the curve at \(P\) has the equation $$x + 3y + 4 = 0.$$ [3]

The tangent to the curve at \(P\) meets the curve again at the point \(Q\).

1. Find the coordinates of \(Q\). [7]

The curve \(C\) has parametric equations $$x = 15t - t^3, \quad y = 3 - 2t^2.$$ Find the values of \(t\) at the points where the normal to \(C\) at \((14, 1)\) cuts \(C\) again. [11]