Why do x=ct, y=\frac ct satisfy xy=c^2?

ct\times\frac ct=c^2; the t cancels.

What does replacing t by \frac1t do to \left(ct,\frac ct\right)?

Swaps the coordinates: \left(\frac ct,ct\right).

Area of the rectangle from O to \left(ct,\frac ct\right) with sides on the axes?

c^2, whatever the value of t (t0).

The rectangular hyperbolaEdexcel International A Level Further Maths: Flashcards

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Cartesian equation of a rectangular hyperbola?

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Cartesian equation of a rectangular hyperbola?: $xy=c^2$ (with $c>0$ )
Parametric equations of $xy=c^2$ ?: $x=ct,\ y=\frac ct,\ t\neq0$
What is the general point on $xy=c^2$ ?: $\left(ct,\frac ct\right)$
Why do $x=ct$ , $y=\frac ct$ satisfy $xy=c^2$ ?: $ct\times\frac ct=c^2$ ; the $t$ cancels.
Cartesian equation of $x=4t,\ y=\frac4t$ ?: $xy=16$
Parametric form of $xy=49$ ?: $x=7t,\ y=\frac7t$ (here $c=7$ )
Point on $xy=36$ with $t=3$ ?: $(18,2)$
Point on $xy=36$ with $x=-9$ : $t$ and $y$ ?: $t=-\frac32$ , $y=-4$
Which quadrants hold the branches of $xy=c^2$ ?: First ( $t>0$ ) and third ( $t<0$ ).
What are the asymptotes of $xy=c^2$ ?: The $x$ -axis and the $y$ -axis.
What does replacing $t$ by $\frac1t$ do to $\left(ct,\frac ct\right)$ ?: Swaps the coordinates: $\left(\frac ct,ct\right)$ .
Area of the rectangle from $O$ to $\left(ct,\frac ct\right)$ with sides on the axes?: $c^2$ , whatever the value of $t$ ( $t>0$ ).
Gradient of the chord joining the points with parameters $p$ and $q$ ?: $-\frac{1}{pq}$

Exam questions on The rectangular hyperbola

A rectangular hyperbola $H$ has equation $xy=36$ .
The point $Q$ lies on $H$ and has $x$ -coordinate $-9$ . Find the value of $t$ at $Q$ and the $y$ -coordinate of $Q$ .2 marks
A curve $C$ has parametric equations $x=5t$ , $y=\frac{5}{t}$ , where $t\neq0$ .
Find the coordinates of the points where $C$ meets the line $y=x$ .2 marks
The rectangular hyperbola $H$ has parametric equations $x=2t$ , $y=\frac{2}{t}$ , $t\neq0$ . The point $A$ has parameter $t=a$ and the point $B$ has parameter $t=2a$ , where $a>0$ .
Write down the coordinates of $A$ and $B$ in terms of $a$ , and hence show that the gradient of the chord $AB$ is $-\frac{1}{2a^2}$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).