Parabola focus and directrix properties

Edexcel F1 2022 January Q3

5 marks Standard +0.8

3. The parabola $C$ has equation $y ^ { 2 } = 18 x$ The point $S$ is the focus of $C$

Write down the coordinates of $S$ The point $P$, with $y > 0$, lies on $C$ The shortest distance from $P$ to the directrix of $C$ is 9 units.
Determine the exact perimeter of the triangle $O P S$, where $O$ is the origin. Give your answer in simplest form.

Edexcel F1 2015 June Q4

6 marks Standard +0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{04f06398-ff29-4690-a6fe-825d089fba39-05_663_665_228_644} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 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 $Q$ lies on the directrix of $C$. The point $P$ lies on $C$ where $y > 0$ and the line segment $Q P$ is parallel to the $x$-axis. Given that the length of $P S$ is 13

write down the length of $P Q$. Given that the point $P$ has $x$ coordinate 9
find
the value of $a$,
the area of triangle $P S Q$.

Edexcel F1 2018 June Q3

5 marks Standard +0.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{b3a6bed4-2d9c-48a3-8831-efb5ba09baa4-08_536_533_221_708} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows the parabola $C$ which has cartesian equation $y ^ { 2 } = 6 x$. The point $S$ is the focus of $C$.

Find the coordinates of the point $S$. The point $P$ lies on the parabola $C$, and the point $Q$ lies on the directrix of $C$. $P Q$ is parallel to the $x$-axis with distance $P Q = 14$
State the distance $S P$. Given that the point $P$ is above the $x$-axis,
find the exact coordinates of $P$.

Edexcel F1 Specimen Q3

7 marks Standard +0.3

3. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{fa5a23b5-d52c-4bae-97c7-2eb7220a3dc4-04_736_659_299_660} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the parabola $C$ with equation $y ^ { 2 } = 8 x$.
The point $P$ lies on $C$, where $y > 0$, and the point $Q$ lies on $C$, where $y < 0$ The line segment $P Q$ is parallel to the $y$-axis. Given that the distance $P Q$ is 12 ,

write down the $y$ coordinate of $P$,
find the $x$ coordinate of $P$. Figure 1 shows the point $S$ which is the focus of $C$. The line $l$ passes through the point $P$ and the point $S$.
Find an equation for $l$ in the form $a x + b y + c = 0$, where $a$, $b$ and $c$ are integers.

Edexcel FP1 2010 January Q4

6 marks Standard +0.3

4. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{cfad960a-f56a-4471-b4ad-92ab670d8121-05_791_874_265_518} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of part of the parabola with equation $y ^ { 2 } = 12 x$.
The point $P$ on the parabola has $x$-coordinate $\frac { 1 } { 3 }$.
The point $S$ is the focus of the parabola.

Write down the coordinates of $S$. The points $A$ and $B$ lie on the directrix of the parabola.
The point $A$ is on the $x$-axis and the $y$-coordinate of $B$ is positive. Given that $A B P S$ is a trapezium,
calculate the perimeter of $A B P S$.

Edexcel FP1 2011 January Q6

8 marks Standard +0.3

6. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{d20fa710-2d91-4ac2-adbc-46ccdcb93380-07_789_791_228_566} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the parabola $C$ with equation $y ^ { 2 } = 36 x$. The point $S$ is the focus of $C$.

Find the coordinates of $S$.
Write down the equation of the directrix of $C$. Figure 1 shows the point $P$ which lies on $C$, where $y > 0$, and the point $Q$ which lies on the directrix of $C$. The line segment $Q P$ is parallel to the $x$-axis. Given that the distance $P S$ is 25 ,
write down the distance $Q P$,
find the coordinates of $P$,
find the area of the trapezium $O S P Q$.

Edexcel FP1 2012 June Q5

7 marks Standard +0.3

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{5e512646-962b-424b-af5f-a6c6b332e0c9-06_732_654_258_646} \captionsetup{labelformat=empty} \caption{Figure 1}

\end{figure} Figure 1 shows a sketch of the parabola $C$ with equation $y ^ { 2 } = 8 x$. The point $P$ lies on $C$, where $y > 0$, and the point $Q$ lies on $C$, where $y < 0$ The line segment $P Q$ is parallel to the $y$-axis. Given that the distance $P Q$ is 12 ,

write down the $y$-coordinate of $P$,
find the $x$-coordinate of $P$. Figure 1 shows the point $S$ which is the focus of $C$.
The line $l$ passes through the point $P$ and the point $S$.
Find an equation for $l$ in the form $a x + b y + c = 0$, where $a$, $b$ and $c$ are integers.

Edexcel FP1 2018 June Q7

8 marks Standard +0.8

The parabola $C$ has equation $y ^ { 2 } = 4 a x$, where $a$ is a positive constant. The point $S$ is the focus of $C$.

The straight line $l$ passes through the point $S$ and meets the directrix of $C$ at the point $D$.
Given that the $y$ coordinate of $D$ is $\frac { 24 a } { 5 }$,

show that an equation of the line $l$ is $$12 x + 5 y = 12 a$$ The point $P \left( a k ^ { 2 } , 2 a k \right)$, where $k$ is a positive constant, lies on the parabola $C$.
Given that the line segment $S P$ is perpendicular to $l$,
find, in terms of $a$, the coordinates of the point $P$.

Edexcel FP1 Q2

3 marks Moderate -0.5

2. A point $P$ with coordinates $( x , y )$ moves so that its distance from the point $( - 3,0 )$ is equal to its distance from the line $x = 3$. Find a cartesian equation for the locus of $P$.

Edexcel FP1 AS 2022 June Q4

9 marks Challenging +1.2

The parabola $C$ has equation $y ^ { 2 } = 10 x$

The point $F$ is the focus of $C$.

Write down the coordinates of $F$. The point $P$ on $C$ has $y$ coordinate $q$, where $q > 0$
Show that an equation for the tangent to $C$ at $P$ is given by $$10 x - 2 q y + q ^ { 2 } = 0$$ The tangent to $C$ at $P$ intersects the directrix of $C$ at the point $A$.
The point $B$ lies on the directrix such that $P B$ is parallel to the $x$-axis.
Show that the point of intersection of the diagonals of quadrilateral $P B A F$ always lies on the $y$-axis.

Edexcel FP1 AS 2024 June Q5

9 marks Challenging +1.2

The parabola $C$ has equation $y ^ { 2 } = 16 x$

The point $P$ on $C$ has $y$ coordinate $p$, where $p$ is a positive constant.

Show that an equation of the tangent to $C$ at $P$ is given by $$2 p y = 16 x + p ^ { 2 }$$ $$\left[ Y \text { ou may quote without proof that for the general parabola } y ^ { 2 } = 4 a x , \frac { d y } { d x } = \frac { 2 a } { y } \right]$$
Write down the equation of the directrix of $C$. The line $l$ is the reflection of the tangent to $C$ at $P$ in the directrix of $C$.
Given that $l$ passes through the focus of $C$,
determine the exact value of $p$.

Edexcel FP1 Specimen Q5

9 marks Challenging +1.2

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

The line $O P$ is perpendicular to the line $O Q$, where $O$ is the origin.
Prove that $p ^ { 2 } = 2$