Parabola tangent intersection problems

Edexcel F1 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 tangent to $C$ at $P$ is $$p y = x + a p ^ { 2 }$$ The tangent to $C$ at the point $P$ intersects the directrix of $C$ at the point $B$ and intersects the $x$-axis at the point $D$. Given that the $y$-coordinate of $B$ is $\frac { 5 } { 6 } a$ and $p > 0$,
find, in terms of $a$, the $x$-coordinate of $D$. Given that $O$ is the origin,
find, in terms of $a$, the area of the triangle $O P D$, giving your answer in its simplest form.

Edexcel F1 2017 June Q8

11 marks Standard +0.3

8. The parabola $C$ has cartesian equation $y ^ { 2 } = 36 x$. The point $P \left( 9 p ^ { 2 } , 18 p \right)$, where $p$ is a positive constant, lies on $C$.

Using calculus, show that an equation of the tangent to $C$ at $P$ is $$p y - x = 9 p ^ { 2 }$$ This tangent cuts the directrix of $C$ at the point $A ( - a , 6 )$, where $a$ is a constant.
Write down the value of $a$.
Find the exact value of $p$.
Hence find the exact coordinates of the point $P$, giving each coordinate as a simplified surd.

Edexcel F1 2021 October Q8

10 marks Standard +0.8

The parabola $C$ has equation $y ^ { 2 } = 20 x$

The point $P$ on $C$ has coordinates ( $5 p ^ { 2 } , 10 p$ ) where $p$ is a non-zero constant.

Use calculus to show that the tangent to $C$ at $P$ has equation $$p y - x = 5 p ^ { 2 }$$ The tangent to $C$ at $P$ meets the $y$-axis at the point $A$.
Write down the coordinates of $A$. The point $S$ is the focus of $C$.
Write down the coordinates of $S$. The straight line $l _ { 1 }$ passes through $A$ and $S$.
The straight line $l _ { 2 }$ passes through $O$ and $P$, where $O$ is the origin. Given that $l _ { 1 }$ and $l _ { 2 }$ intersect at the point $B$,
show that the coordinates of $B$ satisfy the equation $$2 x ^ { 2 } + y ^ { 2 } = 10 x$$

Edexcel FP1 Q8

12 marks Standard +0.8

8. A parabola has equation $y ^ { 2 } = 4 a x , a > 0$. The point $Q \left( a q ^ { 2 } , 2 a q \right)$ lies on the parabola.

Show that an equation of the tangent to the parabola at $Q$ is $$y q = x + a q ^ { 2 }$$ This tangent meets the $y$-axis at the point $R$.
Find an equation of the line $l$ which passes through $R$ and is perpendicular to the tangent at $Q$.
Show that $l$ passes through the focus of the parabola.
Find the coordinates of the point where $l$ meets the directrix of the parabola.

Edexcel FP1 Q8

Standard +0.3

8. A parabola has equation $y ^ { 2 } = 4 a x , a > 0$. The point $Q \left( a q ^ { 2 } , 2 a q \right)$ lies on the parabola.

Show that an equation of the tangent to the parabola at $Q$ is $$y q = x + a q ^ { 2 }$$ This tangent meets the $y$-axis at the point $R$.
Find an equation of the line $l$ which passes through $R$ and is perpendicular to the tangent at $Q$.
Show that $l$ passes through the focus of the parabola.
Find the coordinates of the point where $I$ meets the directrix of the parabola.

Edexcel FP1 2009 January Q8

10 marks Standard +0.8

8. A parabola has equation $y ^ { 2 } = 4 a x , a > 0$. The point $Q \left( a q ^ { 2 } , 2 a q \right)$ lies on the parabola.

Show that an equation of the tangent to the parabola at $Q$ is $$y q = x + a q ^ { 2 } .$$ This tangent meets the $y$-axis at the point $R$.
Find an equation of the line $l$ which passes through $R$ and is perpendicular to the tangent at $Q$.
Show that $l$ passes through the focus of the parabola.
Find the coordinates of the point where $l$ meets the directrix of the parabola.

Edexcel FP1 2017 June Q7

10 marks Standard +0.8

7. The parabola $C$ has equation $y ^ { 2 } = 4 a x$, where $a$ is a constant and $a > 0$ The point $Q \left( a q ^ { 2 } , 2 a q \right) , q > 0$, lies on the parabola $C$.

Show that an equation of the tangent to $C$ at $Q$ is $$q y = x + a q ^ { 2 }$$ The tangent to $C$ at the point $Q$ meets the $x$-axis at the point $X \left( - \frac { 1 } { 4 } a , 0 \right)$ and meets the directrix of $C$ at the point $D$.
Find, in terms of $a$, the coordinates of $D$. Given that the point $F$ is the focus of the parabola $C$,
find the area, in terms of $a$, of the triangle $F X D$, giving your answer in its simplest form.