Parabola normal equation derivation

1. The parabola \(C\) has equation \(y ^ { 2 } = \frac { 4 } { 3 } x\)

The point \(P \left( \frac { 1 } { 3 } t ^ { 2 } , \frac { 2 } { 3 } t \right)\), where \(t \neq 0\), lies on \(C\).

1. Use calculus to show that the normal to \(C\) at \(P\) has equation $$3 t x + 3 y = t ^ { 3 } + 2 t$$ The normal to \(C\) at the point where \(t = 9\) meets \(C\) again at the point \(Q\).
2. Determine the exact coordinates of \(Q\).

3. A parabola \(C\) has cartesian equation \(y ^ { 2 } = 16 x\). The point \(P \left( 4 t ^ { 2 } , 8 t \right)\) is a general point on \(C\).

1. Write down the coordinates of the focus \(F\) and the equation of the directrix of \(C\).
2. Show that the equation of the normal to \(C\) at \(P\) is \(y + t x = 8 t + 4 t ^ { 3 }\).

6. The parabola \(C\) has equation \(y ^ { 2 } = 16 x\).

1. Verify that the point \(P \left( 4 t ^ { 2 } , 8 t \right)\) is a general point on \(C\).
2. Write down the coordinates of the focus \(S\) of \(C\).
3. Show that the normal to \(C\) at \(P\) has equation $$y + t x = 8 t + 4 t ^ { 3 }$$ The normal to \(C\) at \(P\) meets the \(x\)-axis at the point \(N\).
4. Find the area of triangle \(P S N\) in terms of \(t\), giving your answer in its simplest form.

7. The parabola \(C\) has cartesian equation \(y ^ { 2 } = 4 a x , a > 0\) The points \(P \left( a p ^ { 2 } , 2 a p \right)\) and \(P ^ { \prime } \left( a p ^ { 2 } , - 2 a p \right)\) lie on \(C\).

1. Show that an equation of the normal to \(C\) at the point \(P\) is $$y + p x = 2 a p + a p ^ { 3 }$$
2. Write down an equation of the normal to \(C\) at the point \(P ^ { \prime }\). The normal to \(C\) at \(P\) meets the normal to \(C\) at \(P ^ { \prime }\) at the point \(Q\).
3. Find, in terms of \(a\) and \(p\), the coordinates of \(Q\). Given that \(S\) is the focus of the parabola,
4. find the area of the quadrilateral \(S P Q P ^ { \prime }\).

6. The parabola \(C\) has Cartesian equation \(y ^ { 2 } = 8 x\) The point \(P \left( 2 p ^ { 2 } , 4 p \right)\) and the point \(Q \left( 2 q ^ { 2 } , 4 q \right)\), where \(p , q \neq 0 , p \neq q\), are points on \(C\).

1. Show that an equation of the normal to \(C\) at \(P\) is $$y + p x = 2 p ^ { 3 } + 4 p$$
2. Write down an equation of the normal to \(C\) at \(Q\) The normal to \(C\) at \(P\) and the normal to \(C\) at \(Q\) meet at the point \(N\)
3. Show that \(N\) has coordinates $$\left( 2 \left( p ^ { 2 } + p q + q ^ { 2 } + 2 \right) , - 2 p q ( p + q ) \right)$$ The line \(O N\), where \(O\) is the origin, is perpendicular to the line \(P Q\)
4. Find the value of \(( p + q ) ^ { 2 } - 3 p q\)

1. The point \(P \left( a p ^ { 2 } , 2 a p \right)\), where \(a\) is a positive constant, lies on the parabola with equation

$$y ^ { 2 } = 4 a x$$ The normal to the parabola at \(P\) meets the parabola again at the point \(Q \left( a q ^ { 2 } , 2 a q \right)\)

1. Show that $$q = \frac { - p ^ { 2 } - 2 } { p }$$
2. Hence show that $$P Q ^ { 2 } = \frac { k a ^ { 2 } } { p ^ { 4 } } \left( p ^ { 2 } + 1 \right) ^ { n }$$ where \(k\) and \(n\) are integers to be determined.

The parabola \(C\) has equation \(y^2 = 4ax\), where \(a\) is a constant.

1. Show that an equation for the normal to \(C\) at the point \(P(ap^2, 2ap)\) is \(y + px = 2ap + ap^3\). [4]

The normals to \(C\) at the points \(P(ap^2, 2ap)\) and \(Q(aq^2, 2aq)\), \(p \neq q\), meet at the point \(R\).

1. Find, in terms of \(a\), \(p\) and \(q\), the coordinates of \(R\). [5]