Parabola tangent equation derivation

A question is this type if and only if it asks to show or derive the equation of a tangent to a parabola y²=4ax at a general point, typically using calculus.

5 questions · Standard +0.5

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Edexcel F1 2018 January Q6

12 marks Standard +0.3

The parabola $C$ has equation $y ^ { 2 } = 32 x$ and the point $S$ is the focus of this parabola. The point $P ( 2,8 )$ lies on $C$ and the point $T$ lies on the directrix of $C$. The line segment $P T$ is parallel to the $x$-axis.
1. Write down the coordinates of $S$.
2. Find the length of $P T$.
3. Using calculus, show that the tangent to $C$ at the point $P$ has equation
$$y = 2 x + 4$$ The hyperbola $H$ has equation $x y = 4$. The tangent to $C$ at $P$ meets $H$ at the points $L$ and $M$.
Find the exact coordinates of the points $L$ and $M$, giving your answers in their simplest form.

Edexcel FP1 2011 June Q8

10 marks Moderate -0.3

8. The parabola $C$ has equation $y ^ { 2 } = 48 x$. The point $P \left( 12 t ^ { 2 } , 24 t \right)$ is a general point on $C$.

Find the equation of the directrix of $C$.
Show that the equation of the tangent to $C$ at $P \left( 12 t ^ { 2 } , 24 t \right)$ is $$x - t y + 12 t ^ { 2 } = 0$$ The tangent to $C$ at the point $( 3,12 )$ meets the directrix of $C$ at the point $X$.
Find the coordinates of $X$.

Edexcel FP1 2013 June Q7

8 marks Standard +0.8

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

Show that the equation of the tangent to $C$ at $P \left( a t ^ { 2 } , 2 a t \right)$ is $$t y = x + a t ^ { 2 }$$ The tangent to $C$ at $P$ meets the $y$-axis at a point $Q$.
Find the coordinates of $Q$. Given that the point $S$ is the focus of $C$,
show that $P Q$ is perpendicular to $S Q$.

Edexcel FP1 2013 June Q6

13 marks Standard +0.8

6. A curve $C$ is in the form of a parabola with equation $y ^ { 2 } = 4 x$.
$P \left( p ^ { 2 } , 2 p \right)$ and $Q \left( q ^ { 2 } , 2 q \right)$ are points on $C$ where $p > q$.

Find an equation of the tangent to $C$ at $P$.
(5)
The tangent at $P$ and the tangent at $Q$ are perpendicular and intersect at the point $R ( - 1,2 )$.
1. Find the exact value of $p$ and the exact value of $q$.
2. Find the area of the triangle $P Q R$.

Edexcel FP1 Specimen Q7

12 marks Standard +0.8

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

Find the value of $a$.
Show that the equation for the tangent to $C$ at the point $\left( 4 t ^ { 2 } , 8 t \right)$ is $$y t = x + 4 t ^ { 2 } .$$ The tangent to $C$ at the point $A$ meets the tangent to $C$ at the point $B$ on the directrix of $C$ when $y = 15$.
Find the coordinates of $A$ and the coordinates of $B$.