Parabola normal intersection problems

Questions involving normals to parabolas y²=4ax intersecting axes, directrix, or other lines, and computing related quantities like distances or coordinates.

6 questions · Challenging +1.3

1.03g Parametric equations: of curves and conversion to cartesian 1.07m Tangents and normals: gradient and equations

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Edexcel F1 2015 January Q4

14 marks Challenging +1.2

4. The parabola $C$ has cartesian equation $y ^ { 2 } = 12 x$ The point $P \left( 3 p ^ { 2 } , 6 p \right)$ lies on $C$, where $p \neq 0$

Show that the equation of the normal to the curve $C$ at the point $P$ is $$y + p x = 6 p + 3 p ^ { 3 }$$ This normal crosses the curve $C$ again at the point $Q$.
Given that $p = 2$ and that $S$ is the focus of the parabola, find
the coordinates of the point $Q$,
the area of the triangle $P Q S$.

Edexcel F1 2017 January Q8

12 marks Hard +2.3

8. The parabola $C$ has equation $y ^ { 2 } = 4 a x$, where $a$ is a positive constant. The point $P \left( a t ^ { 2 } , 2 a t \right)$ lies on $C$.

Using calculus, show that the normal to $C$ at $P$ has equation $$y + t x = a t ^ { 3 } + 2 a t$$ The point $S$ is the focus of the parabola $C$.
The point $B$ lies on the positive $x$-axis and $O B = 5 O S$, where $O$ is the origin.
Write down, in terms of $a$, the coordinates of the point $B$. A circle has centre $B$ and touches the parabola $C$ at two distinct points $Q$ and $R$. Given that $t \neq 0$,
find the coordinates of the points $Q$ and $R$.
Hence find, in terms of $a$, the area of triangle $B Q R$.

Edexcel F1 2022 June Q6

11 marks Challenging +1.3

The parabola $C$ has equation $y ^ { 2 } = 36 x$

The point $P \left( 9 t ^ { 2 } , 18 t \right)$, where $t \neq 0$, lies on $C$

Use calculus to show that the normal to $C$ at $P$ has equation $$y + t x = 9 t ^ { 3 } + 18 t$$
Hence find the equations of the two normals to $C$ which pass through the point (54, 0), giving your answers in the form $y = p x + q$ where $p$ and $q$ are constants to be determined. Given that
- the normals found in part (b) intersect the directrix of $C$ at the points $A$ and $B$
- the point $F$ is the focus of $C$
- determine the area of triangle $A F B$

Edexcel FP1 2013 January Q9

9 marks Standard +0.8

9. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{7833e9c0-4a73-4ac6-8a77-51a5489e0614-10_624_716_210_614} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the parabola with equation $y ^ { 2 } = 36 x$.
The point $P ( 4,12 )$ lies on the parabola.

Find an equation for the normal to the parabola at $P$. This normal meets the $x$-axis at the point $N$ and $S$ is the focus of the parabola, as shown in Figure 1.
Find the area of triangle $P S N$.

Edexcel FP1 2014 January Q8

12 marks Standard +0.8

8. The parabola $C$ has equation $y ^ { 2 } = 4 a x$, where $a$ is a positive constant. The point $P \left( a p ^ { 2 } , 2 a p \right)$ lies on the parabola $C$.

Show that an equation of the normal to $C$ at $P$ is $$y + p x = a p ^ { 3 } + 2 a p$$ The normal to $C$ at the point $P$ meets the $x$-axis at the point $( 6 a , 0 )$ and meets the directrix of $C$ at the point $D$. Given that $p > 0$,
find, in terms of $a$, the coordinates of $D$. Given also that the directrix of $C$ cuts the $x$-axis at the point $X$,
find, in terms of $a$, the area of the triangle XPD, giving your answer in its simplest form.

Edexcel FP1 Q39

10 marks Challenging +1.2

The points $P(ap^2, 2ap)$ and $Q(aq^2, 2aq)$, $p \neq q$, lie on the parabola $C$ with equation $y^2 = 4ax$, where $a$ is a constant.

Show that an equation for the chord $PQ$ is $(p + q) y = 2(x + apq)$ . [3]

The normals to $C$ at $P$ and $Q$ meet at the point $R$.

Show that the coordinates of $R$ are $(a(p^2 + q^2 + pq + 2), -apq(p + q) )$. [7]