Parabola area calculations

Edexcel F1 2016 January Q8

8 marks Challenging +1.2

8. The parabola $P$ has equation $y ^ { 2 } = 4 a x$, where $a$ is a positive constant. The point $S$ is the focus of $P$. The point $B$, which does not lie on the parabola, has coordinates ( $q , r$ ) where $q$ and $r$ are positive constants and $q > a$. The line $l$ passes through $B$ and $S$.

Show that an equation of the line $l$ is $$( q - a ) y = r ( x - a )$$ The line $l$ intersects the directrix of $P$ at the point $C$. Given that the area of triangle $O C S$ is three times the area of triangle $O B S$, where $O$ is the origin,
show that the area of triangle $O B C$ is $\frac { 6 } { 5 } \mathrm { qr }$

Edexcel F1 2023 January Q8

11 marks Challenging +1.8

A parabola $C$ has equation $y ^ { 2 } = 4 a x$ where $a$ is a positive constant.

The point $S$ is the focus of $C$
The line $l _ { 1 }$ with equation $y = k$ where $k$ is a positive constant, intersects $C$ at the point $P$

Show that $$P S = \frac { k ^ { 2 } + 4 a ^ { 2 } } { 4 a }$$ The line $l _ { 2 }$ passes through $P$ and intersects the directrix of $C$ on the $x$-axis.
The line $l _ { 2 }$ intersects the $y$-axis at the point $A$
Show that the $y$ coordinate of $A$ is $\frac { 4 a ^ { 2 } k } { k ^ { 2 } + 4 a ^ { 2 } }$ The line $l _ { 1 }$ intersects the directrix of $C$ at the point $B$
Given that the areas of triangles $B P A$ and $O S P$, where $O$ is the origin, satisfy the ratio $$\text { area } B P A \text { : area } O S P = 4 k ^ { 2 } : 1$$
determine the exact value of $a$

Edexcel F1 2024 January Q3

6 marks Standard +0.8

The hyperbola $H$ has equation $x y = c ^ { 2 }$ where $c$ is a positive constant.

The point $P \left( c t , \frac { c } { t } \right)$, where $t > 0$, lies on $H$.
The tangent to $H$ at $P$ meets the $x$-axis at the point $A$ and meets the $y$-axis at the point $B$.

Determine, in terms of $c$ and $t$,
1. the coordinates of $A$,
2. the coordinates of $B$. Given that the area of triangle $A O B$, where $O$ is the origin, is 90 square units,
determine the value of $c$, giving your answer as a simplified surd.

Edexcel F1 2016 June Q2

5 marks Standard +0.8

2. A parabola $P$ has cartesian equation $y ^ { 2 } = 28 x$. The point $S$ is the focus of the parabola $P$.

Write down the coordinates of the point $S$. Points $A$ and $B$ lie on the parabola $P$. The line $A B$ is parallel to the directrix of $P$ and cuts the $x$-axis at the midpoint of $O S$, where $O$ is the origin.
Find the exact area of triangle $A B S$.
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Edexcel F1 2020 June Q7

10 marks Standard +0.8

7. The parabola $C$ has equation $y ^ { 2 } = 4 a x$, where $a$ is a positive constant. The line $l$ with equation $3 x - 4 y + 48 = 0$ is a tangent to $C$ at the point $P$.

Show that $a = 9$
Hence determine the coordinates of $P$. Given that the point $S$ is the focus of $C$ and that the line $l$ crosses the directrix of $C$ at the point $A$,
determine the exact area of triangle $P S A$.
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Edexcel F1 2018 Specimen Q2

5 marks Standard +0.8

A parabola $P$ has cartesian equation $y ^ { 2 } = 28 x$. The point $S$ is the focus of the parabola $P$.
1. Write down the coordinates of the point $S$.
Points $A$ and $B$ lie on the parabola $P$. The line $A B$ is parallel to the directrix of $P$ and cuts the $x$-axis at the midpoint of $O S$, where $O$ is the origin.
Find the exact area of triangle $A B S$.

Edexcel FP1 2012 June Q8

8 marks Standard +0.8

8. The rectangular hyperbola $H$ has equation $x y = c ^ { 2 }$, where $c$ is a positive constant. The point $P \left( c t , \frac { c } { t } \right) , t \neq 0$, is a general point on $H$.

Show that an equation for the tangent to $H$ at $P$ is $$x + t ^ { 2 } y = 2 c t$$ The tangent to $H$ at the point $P$ meets the $x$-axis at the point $A$ and the $y$-axis at the point $B$. Given that the area of the triangle $O A B$, where $O$ is the origin, is 36 ,
find the exact value of $c$, expressing your answer in the form $k \sqrt { } 2$, where $k$ is an integer.

Edexcel FP1 AS 2019 June Q5

10 marks Challenging +1.2

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{6b9c61ac-23ec-4346-933f-cf00a2e63695-12_744_697_294_683} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Figure 2 shows a sketch of part of the rectangular hyperbola $H$ with equation $$x y = c ^ { 2 } \quad x > 0$$ where $c$ is a positive constant.
The point $P \left( c t , \frac { c } { t } \right)$ lies on $H$.
The line $l$ is the tangent to $H$ at the point $P$.
The line $l$ crosses the $x$-axis at the point $A$ and crosses the $y$-axis at the point $B$.
The region $R$, shown shaded in Figure 2, is bounded by the $x$-axis, the $y$-axis and the line $l$. Given that the length $O B$ is twice the length of $O A$, where $O$ is the origin, and that the area of $R$ is 32 , find the exact coordinates of the point $P$.

Edexcel FP1 AS Specimen Q5

12 marks Standard +0.8

5. \begin{figure}[h]

\includegraphics[alt={},max width=\textwidth]{ff1fc9b0-6514-44e0-a2a3-46aa6411ce10-10_965_853_212_621} \captionsetup{labelformat=empty} \caption{Figure 2}

\end{figure} Diagram not drawn to scale $$\left[ Y \text { ou may quote without proof that for the general parabola } y ^ { 2 } = 4 a x , \frac { d y } { d x } = \frac { 2 a } { y } \right]$$ The parabola C has equation $\mathrm { y } ^ { 2 } = 16 \mathrm { x }$

Deduce that the point $\mathrm { P } \left( 4 \mathrm { p } ^ { 2 } , 8 \mathrm { p } \right)$ is a general point on C . The line I is the tangent to C at the point P .
Show that an equation for I is $$p y = x + 4 p ^ { 2 }$$ The finite region R , shown shaded in Figure 2, is bounded by the line I , the x -axis and the parabola C.
The line $I$ intersects the directrix of $C$ at the point $B$, where the $y$ coordinate of $B$ is $\frac { 10 } { 3 }$ Given that $\mathrm { p } > 0$
show that the area of R is 36 \section*{Q uestion 5 continued}

Edexcel FP1 2019 June Q4

8 marks Challenging +1.2

The parabola $C$ has equation

$$y ^ { 2 } = 16 x$$ The distinct points $P \left( p ^ { 2 } , 4 p \right)$ and $Q \left( q ^ { 2 } , 4 q \right)$ lie on $C$, where $p \neq 0 , q \neq 0$
The tangent to $C$ at $P$ and the tangent to $C$ at $Q$ meet at the point $R ( - 28,6 )$.
Show that the area of triangle $P Q R$ is 1331

Edexcel FP1 2024 June Q8

8 marks Challenging +1.2

The parabola $P$ has equation $y ^ { 2 } = 4 a x$, where $a$ is a positive constant.

The point $A \left( a t ^ { 2 } , 2 a t \right)$, where $t \neq 0$, lies on $P$.

Use calculus to show that an equation of the tangent to $P$ at $A$ is $$y t = x + a t ^ { 2 }$$ The point $B \left( 2 k ^ { 2 } , 4 k \right)$ and the point $C \left( 2 k ^ { 2 } , - 4 k \right)$, where $k$ is a constant, lie on $P$.
The tangent to $P$ at $B$ and the tangent to $P$ at $C$ intersect at the point $D$.
Given that the area of the triangle $B C D$ is 432
determine the coordinates of $B$ and the coordinates of $C$.