12 A curve has parametric equations \(x = \frac { 1 } { t } , y = 2 t\). The point \(P\) is \(\left( \frac { 1 } { p } , 2 p \right)\).
- 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 \(B\).
- Show that the ratio \(P A : P B\) is \(1 : 2 p ^ { 2 }\).
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