Rectangular hyperbola normal re-intersection

Edexcel F1 2016 January Q6

9 marks Challenging +1.8

6. The rectangular hyperbola $H$ has equation $x y = c ^ { 2 }$, where $c$ is a non-zero constant. The point $P \left( c p , \frac { c } { p } \right)$, where $p \neq 0$, lies on $H$.

Show that the normal to $H$ at $P$ has equation $$y p - p ^ { 3 } x = c \left( 1 - p ^ { 4 } \right)$$ The normal to $H$ at $P$ meets $H$ again at the point $Q$.
Find, in terms of $c$ and $p$, the coordinates of $Q$.

Edexcel F1 2023 January Q6

9 marks Challenging +1.8

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

The rectangular hyperbola $H$ has equation $x y = 20$
The point $P \left( 2 t \sqrt { a } , \frac { 2 \sqrt { a } } { t } \right) , t \neq 0$, where $a$ is a constant, is a general point on $H$

State the value of $a$
Show that the normal to $H$ at the point $P$ has equation $$t y - t ^ { 3 } x - 2 \sqrt { 5 } \left( 1 - t ^ { 4 } \right) = 0$$ The points $A$ and $B$ lie on $H$
The point $A$ has parameter $t = c$ and the point $B$ has parameter $t = - \frac { 1 } { 2 c }$, where $c$ is a constant. The normal to $H$ at $A$ meets $H$ again at $B$
Determine the possible values of $C$

Edexcel F1 2016 June Q6

10 marks Standard +0.8

6. The rectangular hyperbola $H$ has equation $x y = 25$

Verify that, for $t \neq 0$, the point $P \left( 5 t , \frac { 5 } { t } \right)$ is a general point on $H$. The point $A$ on $H$ has parameter $t = \frac { 1 } { 2 }$
Show that the normal to $H$ at the point $A$ has equation $$8 y - 2 x - 75 = 0$$ This normal at $A$ meets $H$ again at the point $B$.
Find the coordinates of $B$.

Edexcel F1 2020 June Q5

9 marks Challenging +1.2

The rectangular hyperbola $H$ has equation $x y = 64$

The point $P \left( 8 p , \frac { 8 } { p } \right)$, where $p \neq 0$, lies on $H$.

Use calculus to show that the normal to $H$ at $P$ has equation $$p ^ { 3 } x - p y = 8 \left( p ^ { 4 } - 1 \right)$$ The normal to $H$ at $P$ meets $H$ again at the point $Q$.
Determine, in terms of $p$, the coordinates of $Q$, giving your answers in simplest form. \includegraphics[max width=\textwidth, alt={}, center]{a3457c24-fbda-413d-b3b2-6be375307318-17_2255_50_314_34}

Edexcel F1 2023 June Q3

7 marks Challenging +1.2

The rectangular hyperbola $H$ has Cartesian equation $x y = 9$

The point $P$ with coordinates $\left( 3 t , \frac { 3 } { t } \right)$, where $t \neq 0$, lies on $H$

Use calculus to determine an equation for the normal to $H$ at the point $P$ Give your answer in the form $t y - t ^ { 3 } x = \mathrm { f } ( t )$ Given that $t = 2$
determine the coordinates of the point where the normal meets $H$ again. Give your answer in simplest form.

Edexcel F1 2018 Specimen Q6

10 marks Standard +0.3

The rectangular hyperbola $H$ has equation $x y = 25$
1. Verify that, for $t \neq 0$, the point $P \left( 5 t , \frac { 5 } { t } \right)$ is a general point on $H$.
The point $A$ on $H$ has parameter $t = \frac { 1 } { 2 }$
Show that the normal to $H$ at the point $A$ has equation $$8 y - 2 x - 75 = 0$$ This normal at $A$ meets $H$ again at the point $B$.
Find the coordinates of $B$.
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Edexcel FP1 2010 June Q8

11 marks Challenging +1.2

8. The rectangular hyperbola $H$ has equation $x y = c ^ { 2 }$, where c is a positive constant. The point $A$ on $H$ has $x$-coordinate $3 c$.

Write down the $y$-coordinate of $A$.
Show that an equation of the normal to $H$ at $A$ is $$3 y = 27 x - 80 c$$ The normal to $H$ at $A$ meets $H$ again at the point $B$.
Find, in terms of $c$, the coordinates of $B$.

Edexcel FP1 2013 June Q4

10 marks Challenging +1.2

4. The hyperbola $H$ has equation $$x y = 3$$ The point $Q ( 1,3 )$ is on $H$.

Find the equation of the normal to $H$ at $Q$ in the form $y = a x + b$, where $a$ and $b$ are constants.
(5) The normal at $Q$ intersects $H$ again at the point $R$.
Find the coordinates of $R$.
(5)

Edexcel FP1 2013 June Q4

9 marks Standard +0.8

4. The rectangular hyperbola $H$ has Cartesian equation $x y = 4$ The point $P \left( 2 t , \frac { 2 } { t } \right)$ lies on $H$, where $t \neq 0$

Show that an equation of the normal to $H$ at the point $P$ is $$t y - t ^ { 3 } x = 2 - 2 t ^ { 4 }$$ The normal to $H$ at the point where $t = - \frac { 1 } { 2 }$ meets $H$ again at the point $Q$.
Find the coordinates of the point $Q$.

Edexcel FP1 2015 June Q5

9 marks Challenging +1.2

5. The rectangular hyperbola $H$ has equation $x y = 9$ The point $A$ on $H$ has coordinates $\left( 6 , \frac { 3 } { 2 } \right)$.

Show that the normal to $H$ at the point $A$ has equation $$2 y - 8 x + 45 = 0$$ The normal at $A$ meets $H$ again at the point $B$.
Find the coordinates of $B$.

Edexcel FP1 Q8

13 marks Standard +0.8

8. The rectangular hyperbola $H$ has equation $x y = c ^ { 2 }$. The point ( $3 t , \frac { 3 } { t }$ ) is a general point on this hyperbola.

Find the value of $c ^ { 2 }$.
Show that an equation of the normal to $H$ at the point ( $3 t , \frac { 3 } { t }$ ) is $$y = t ^ { 2 } x + \left( \frac { 3 } { t } - 3 t ^ { 3 } \right)$$ The point $P$ on $H$ has coordinates (6, 1.5). The tangent to $H$ at $P$ meets the curve again at the point $Q$.
Find the coordinates of the point $Q$.

Edexcel FP1 AS 2018 June Q5

10 marks Challenging +1.2

The rectangular hyperbola $H$ has equation $x y = c ^ { 2 }$, where $c$ is a non-zero constant.

The point $P \left( c p , \frac { c } { p } \right)$, where $p \neq 0$, lies on $H$.

Use calculus to show that an equation of the normal to $H$ at $P$ is $$p ^ { 3 } x - p y + c \left( 1 - p ^ { 4 } \right) = 0$$ The normal to $H$ at the point $P$ meets $H$ again at the point $Q$.
Find the coordinates of the midpoint of $P Q$ in terms of $c$ and $p$, simplifying your answer where possible.