Ellipse locus problems - Conic sections

Edexcel F3 2023 January Q9

13 marks Challenging +1.2

The ellipse $E$ has equation

$$x ^ { 2 } + 9 y ^ { 2 } = 9$$ The foci of $E$ are $F _ { 1 }$ and $F _ { 2 }$

1. Determine the coordinates of $F _ { 1 }$ and the coordinates of $F _ { 2 }$
2. Write down the equation of each of the directrices of $E$ The point $P$ lies on the ellipse.
Show that $\left| P F _ { 1 } \right| + \left| P F _ { 2 } \right| = 6$ The straight line through $P$ with equation $y = 2 x + c$ meets $E$ again at the point $Q$ The point $M$ is the midpoint of $P Q$
Show that as $P$ varies the locus of $M$ is a straight line passing through the origin.

Edexcel F3 2023 June Q6

13 marks Challenging +1.3

The ellipse $E$ has equation $\frac { x ^ { 2 } } { 16 } + \frac { y ^ { 2 } } { 9 } = 1$

The point $P ( 4 \cos \theta , 3 \sin \theta )$ lies on $E$.

Use calculus to show that an equation of the tangent to $E$ at $P$ is $$3 x \cos \theta + 4 y \sin \theta = 12$$
Determine an equation for the normal to $E$ at $P$. The tangent to $E$ at $P$ meets the $x$-axis at the point $A$.
The normal to $E$ at $P$ meets the $y$-axis at the point $B$.
Show that the locus of the midpoint of $A$ and $B$ as $\theta$ varies has equation $$x ^ { 2 } \left( p - q y ^ { 2 } \right) = r$$ where $p , q$ and $r$ are integers to be determined.

Edexcel F3 2021 October Q3

9 marks Challenging +1.8

3. The ellipse $E$ has equation $$\frac { x ^ { 2 } } { 64 } + \frac { y ^ { 2 } } { 36 } = 1$$ The line $l$ is the normal to $E$ at the point $P ( 8 \cos \theta , 6 \sin \theta )$.

Using calculus, show that an equation for $l$ is $$4 x \sin \theta - 3 y \cos \theta = 14 \sin \theta \cos \theta$$ The line $l$ meets the $x$-axis at the point $A$ and meets the $y$-axis at the point $B$.
The point $M$ is the midpoint of $A B$.
Determine a Cartesian equation for the locus of $M$ as $\theta$ varies, giving your answer in the form $a x ^ { 2 } + b y ^ { 2 } = c$ where $a , b$ and $c$ are integers.

Edexcel FP3 2012 June Q6

11 marks Challenging +1.2

The ellipse $E$ has equation

$$\frac { x ^ { 2 } } { a ^ { 2 } } + \frac { y ^ { 2 } } { b ^ { 2 } } = 1$$ The line $l _ { 1 }$ is a tangent to $E$ at the point $P ( a \cos \theta , b \sin \theta )$.

Using calculus, show that an equation for $l _ { 1 }$ is $$\frac { x \cos \theta } { a } + \frac { y \sin \theta } { b } = 1$$ The circle $C$ has equation $$x ^ { 2 } + y ^ { 2 } = a ^ { 2 }$$ The line $l _ { 2 }$ is a tangent to $C$ at the point $Q ( a \cos \theta , a \sin \theta )$.
Find an equation for the line $l _ { 2 }$. Given that $l _ { 1 }$ and $l _ { 2 }$ meet at the point $R$,
find, in terms of $a , b$ and $\theta$, the coordinates of $R$.
Find the locus of $R$, as $\theta$ varies.

Edexcel FP3 2013 June Q3

8 marks Challenging +1.2

The point $P$ lies on the ellipse $E$ with equation

$$\frac { x ^ { 2 } } { 36 } + \frac { y ^ { 2 } } { 9 } = 1$$ $N$ is the foot of the perpendicular from point $P$ to the line $x = 8$
$M$ is the midpoint of $P N$.

Sketch the graph of the ellipse $E$, showing also the line $x = 8$ and a possible position for the line $P N$.
Find an equation of the locus of $M$ as $P$ moves around the ellipse.
Show that this locus is a circle and state its centre and radius.

Edexcel FP3 2015 June Q8

14 marks Challenging +1.2

The ellipse $E$ has equation $x ^ { 2 } + 4 y ^ { 2 } = 4$
1. (i) Find the coordinates of the foci, $F _ { 1 }$ and $F _ { 2 }$, of $E$.
  (ii) Write down the equations of the directrices of $E$.
2. Given that the point $P$ lies on the ellipse, show that
$$\left| P F _ { 1 } \right| + \left| P F _ { 2 } \right| = 4$$ A chord of an ellipse is a line segment joining two points on the ellipse.
The set of midpoints of the parallel chords of $E$ with gradient $m$, where $m$ is a constant, lie on a straight line $l$.
Find an equation of $l$.

Edexcel FP1 AS 2020 June Q4

7 marks Standard +0.8

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{1e5324f5-a9bc-4041-bfbb-cb940417ea63-12_611_608_274_715} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of the parabola $C$ with equation $y ^ { 2 } = 4 a x$, where $a$ is a positive constant. The point $S$ is the focus of $C$ and the point $P \left( a p ^ { 2 } \right.$, 2ap) lies on $C$ where $p > 0$

Write down the coordinates of $S$.
Write down the length of SP in terms of $a$ and $p$. The point $Q \left( a q ^ { 2 } , 2 a q \right)$, where $p \neq q$, also lies on $C$.
The point $M$ is the midpoint of $P Q$.
Given that $p q = - 1$
prove that, as $P$ varies, the locus of $M$ has equation $$y ^ { 2 } = 2 a ( x - a )$$

Edexcel FP1 2020 June Q7

14 marks Challenging +1.8

The points $P \left( 9 p ^ { 2 } , 18 p \right)$ and $Q \left( 9 q ^ { 2 } , 18 q \right) , p \neq q$, lie on the parabola $C$ with equation

$$y ^ { 2 } = 36 x$$ The line $l$ passes through the points $P$ and $Q$

Show that an equation for the line $l$ is $$( p + q ) y = 2 ( x + 9 p q )$$ The normal to $C$ at $P$ and the normal to $C$ at $Q$ meet at the point $A$.
Show that the coordinates of $A$ are $$\left( 9 \left( p ^ { 2 } + q ^ { 2 } + p q + 2 \right) , - 9 p q ( p + q ) \right)$$ Given that the points $P$ and $Q$ vary such that $l$ always passes through the point $( 12,0 )$
find, in the form $y ^ { 2 } = \mathrm { f } ( x )$, an equation for the locus of $A$, giving $\mathrm { f } ( x )$ in simplest form.

Edexcel FP1 Specimen Q7

8 marks Challenging +1.8

$P$ and $Q$ are two distinct points on the ellipse described by the equation $x ^ { 2 } + 4 y ^ { 2 } = 4$

The line $l$ passes through the point $P$ and the point $Q$.
The tangent to the ellipse at $P$ and the tangent to the ellipse at $Q$ intersect at the point $( r , s )$.
Show that an equation of the line $l$ is $$4 s y + r x = 4$$