Coordinate Systems (Edexcel IAL Further Maths): Exam Questions

A parabola equation$P$ has cartesian equation $y^{2}=28x$.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 $AB$ is parallel to the directrix of $P$ and cuts the $x$-axis at the midpoint of $OS$, where $O$ is the origin.

Find the exact area of triangle $ABS$.

The rectangular hyperbola $H$ has equation $xy=25$

Verify that, for $t\neq0$ the point $P ( 5 t ,   \frac{5}{t} )$ 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

$8y-2x-75=0$

This normal at $A$meets $H$ again at the point $B$.

Find the coordinates of $B$.

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

The point $P ( c t ,   \frac{c}{t} )$ , where $t>0$ , lies on $H$

Use calculus to show that an equation of the normal to $H$ at $P$ is

$t^{3}x-ty=c(t^{4}-1)$

The parabola $C$ has equation $y^{2}=6x$

The normal to $H$ at the point with coordinates $(8,2)$ meets $C$ at the point $Q$ where $y>0$

Determine the exact coordinates of $Q$

Given that

• the point $R$ is the focus of $C$
• the line $l$ is the directrix of $C$
• the line through $Q$ and $R$ meets $l$ at the point $S$

determine the exact length of $QS$

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

The point $P ( c t ,   \frac{c}{t} )$, 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 $t$,

(i) the coordinates of $A$,

(ii) the coordinates of $B$.

Given that the area of triangle $AOB$, where $O$is the origin, is 90 square units,

determine the value of $c$, giving your answer as a simplified surd.

The parabola $C$ has equation $y^{2}=\frac{4}{3}x$

The point $P ( 1 \frac{1}{3} t^{2} ,   \frac{2}{3} t )$ , where $t\neq0$,lies on $C$.

Use calculus to show that the normal to $C$ at $P$ has equation

$3tx+3y=t^{3}+2t$

The normal to $C$ at the point where $t=9$ meets $C$ again at the point $Q$.

Determine the exact coordinates of $Q$.

The rectangular hyperbola $H$ has Cartesian equation $xy=9$.

The point $P$ with coordinates $( 3 t ,   \frac{3}{t} )$ , where $t\neq0$ , lies on $H$.

Use calculus to determine an equation for the normal to $H$ at the point $P$.

Give your answer in the form $ty-t^{3}x=f(t)$.

Given that $t=2$

determine the coordinates of the point where the normal meets $H$ again.

Give your answer in simplest form.

The point $P(2p^{2},4p)$ lies on the parabola with equation $y^{2}=8x$

Show that the point $Q ( \frac{2}{p^{2}} ,   \frac{- 4}{p} )$ , where $p\neq0$, lies on the parabola.

Show that the chord $PQ$ passes through the focus of the parabola.

The tangent to the parabola at $P$ and the tangent to the parabola at $Q$ meet at the point $R$

Determine, in simplest form, the coordinates of Determine $R$

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 $xy=20$

The point $P ( 2 t \sqrt{a} ,   \frac{2 \sqrt{a} }{t} )$, $t\neq0$, where $a$ is a constant, is a general point on $H$

State the value of $a$

Show that the normal to equation$H$ at the point $P$ has equation

$t y - t^{3} x - 2 \sqrt{5} ( 1 - t^{4} ) = 0$

The points $A$and $B$ lie on $H$

The point $A$has parameter $t=c$ and the point $B$ has parameter $t=-\frac{1}{2c}$ , where $c$ is a constant.

The normal to $H$ at $A$meets $H$ again at $B$

Determine the possible values of $c$

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

The point $S$ is the focus of $C$

equationThe line $l_{1}$ with equation $y=k$ where $k$ is a positive constant, intersects $C$ at the point $P$

equationShow that

$PS=\frac{k^{2}+4a^{2}}{4a}$equation``equation

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 $\frac{4a^{2}k}{k^{2}+4a^{2}}$

The line $l_{1}$ intersects the directrix of $C$ at the point $B$

Given that the areas of triangles $BPA$ and $OSP$ , where $O$ is the origin, satisfy the ratio

area $BPA:$ area $OSP=4k^{2}:1$

determine the exact value of $a$

The parabola $C$ has equation $y^{2}=36x$

The point $P(9t^{2},18t)$, where $t\neq0$, lies on $C$

Use calculus to show that the normal to $C$ at $P$ has equation

$y+tx=9t^{3}+18t$

Hence find the equations of the two normals to $C$ which pass through the point (54, 0), giving your answers in the form $y=px+q$ where $p$ and $q$ are constants to be determined.

Given that

• the normals found in part (b) intersect the directrix of $C$ at the points $A$and $B$
• the point $F$ is the focus of $C$

determine the area of triangle $AFB$

The parabola $C$ has equation

$y^{2}=18x$

The point $S$is the focus of $C$

Write down the coordinates of $S$

Determine the exact perimeter of the triangle $OPS$ , where $O$ is the origin.

Give your answer in simplest form.

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

$xy=36$

The point $P(4,9)$ lies on $H$

Show, using calculus, that the normal to $H$ at $P$ has equation

$4x-9y+65=0$

equationThe normal to $H$ at $P$ crosses $H$ again at the point $Q$

Determine an equation for the tangent to $H$ at $Q$ , giving your answer in the form $y=mx+c$ where $m$and $c$ are rational constants.

The curve $H$ has equation

$xy=a^{2}x>0$

where $a$ is a positive constant.

The line with equation $y=kx$, where $k$ is a positive constant, intersects $H$ at the point $P$.

Use calculus to determine, in terms of $a$ and $k$, an equation for the tangent to $H$at $P$ .

The tangent to $H$ at $P$ meets the $x$-axis at the point $A$and meets the $y$-axis at the point $B$.

Determine the coordinates of $A$and the coordinates of $B$, giving your answers in terms of $a$ and $k$.

Hence show that the area of triangle $AOB$ , where $o$ is the origin, is independent of $k$.

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

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

Use calculus to show that the tangent to $C$ at $P$ has equation

$py-x=5p^{2}$

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

$2x^{2}+y^{2}=10x$

A rectangular hyperbola $H$ has equation $xy=25$

The point $P ( 5 t ,   \frac{5}{t} )$ , $t\neq0$ , is a general point on $H$.

Show that the equation of the tangent to $H$ at $P$ is $t^{2}y+x=10t$

The distinct points $Q$ and $R$ lie on $H$. The tangent to $H$ at the point $Q$ and the tangent to $H$ at the point $R$ meet at the point $(15,-5)$.

Find the coordinates of the points $Q$ and $R$.

The parabola $C$ has Cartesian equation $y^{2}=8x$

The point $P(2p^{2},4p)$ and the point $Q(2q^{2},4q)$, where $p,q\neq0$ , $p\neq q$, are points on $C$.

Show that an equation of the normal to $C$ at $P$ is

$y+px=2p^{3}+4p$

Write down an equation of the normal to $C$ at $Q$

The normal to $C$ at $P$ and the normal to $C$ at $Q$ meet at the point $N$

Show that $N$ has coordinates

$( 2 ( p^{2} + p q + q^{2} + 2 ) , - 2 p q ( p + q ) )$

The line $ON$, where $O$ is the origin, is perpendicular to the line $PQ$.

Find the value of $(p+q)^{2}-3pq$

The hyperbola $H$ has Cartesian equation $xy=25$

The parabola $P$ has parametric equations $x=10t^{2},y=20t$

The hyperbola $H$ intersects the parabola $P$ at the point $A$.

Use algebra to determine the coordinates of $A$.

The point $B$ with coordinates $(10,20)$ lies on $P$.

Find an equation for the normal to $P$ at $B$ .

Give your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers to be determined.

Use algebra to determine, in simplest form, the exact coordinates of the points where this normal intersects the hyperbola $H$.

The rectangular hyperbola $H$ has equation $xy=64$

equationThe point $P ( 8 p ,   \frac{8}{p} )$, where $p\neq0$ , lies on $H$.

Use calculus to show that the normal to $H$ at $P$ has equation

equation$p^{3}x-py=8(p^{4}-1)$

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.

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

The line $l$ with equation $3x-4y+48=0$ 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 $PSA$.