Perpendicular tangents to a parabola

The points \(P\left(p,p^2\right)\) and \(Q\left(q,q^2\right)\), where \(p\ne q\), lie on the curve \(y=x^2\).

1. Show that the tangent to the curve at \(P\) has equation \(y=2px-p^2\).[2]
2. Show that the tangents at \(P\) and \(Q\) meet at the point \(\left(\dfrac{p+q}{2},\,pq\right)\).[3]
3. Given that the tangents at \(P\) and \(Q\) are perpendicular, show that they meet on the line \(y=-\dfrac14\).[2]
4. Perpendicular tangents at \(P\) and \(Q\) meet at a point with \(x\)-coordinate \(\dfrac34\). Find the exact values of \(p\) and \(q\), given that \(p>q\).[3]