The rectangular hyperbolaEdexcel International A Level Further Maths: Subtopic test

1. 1
   A rectangular hyperbola $H$ has equation $xy=36$.

   (a)

   Which of the following gives a general point on $H$, where $t$ is a non-zero parameter?

   [1 mark]

   • A$\left(6t,\frac{6}{t}\right)$
   • B$\left(36t,\,t\right)$
   • C$\left(6t,\,6t\right)$
   • D$\left(6t,\frac{36}{t}\right)$

   (b)

   Using the parametric form $x=6t$, $y=\frac{6}{t}$, find the coordinates of the point on $H$ with parameter $t=3$.

   [1 mark]

   • A$(2,18)$
   • B$(18,2)$
   • C$(6,3)$
   • D$(9,4)$

   (c)

   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]

   Total for question 1: 4 marks

2. 2
   A curve $C$ has parametric equations $x=5t$, $y=\frac{5}{t}$, where $t\neq0$.

   (a)

   Find the Cartesian equation of $C$.

   [1 mark]

   • A$xy=5$
   • B$xy=10$
   • C$xy=25$
   • D$y=25x$

   (b)

   Which of the following points does NOT lie on $C$?

   [1 mark]

   • A$(-5,-5)$
   • B$(25,1)$
   • C$\left(\frac12,50\right)$
   • D$(2,10)$

   (c)

   Find the coordinates of the points where $C$ meets the line $y=x$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   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 $a>0$.

   (a)

   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]

   (b)

   The midpoint of $AB$ lies on the line $y=x$. Find the value of $a$ and the coordinates of the midpoint.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The rectangular hyperbola $H$ has equation $xy=9$, with $x>0$. The point $P$ lies on $H$ and has parameter $t>0$, where $x=3t$ and $y=\frac3t$. The point $O$ is the origin.

   (a)

   (i) Show that the rectangle with $O$ and $P$ as opposite vertices and sides along the axes has the same area for every value of $t$.
   (ii) Show that $OP^2=9t^2+\frac{9}{t^2}$.
   (iii) Find $OP$ when $t=2$, giving your answer to 3 significant figures.

   [6 marks]

   (b)

   (i) Show that $OP^2=9\left(t-\frac1t\right)^2+18$.
   (ii) Hence explain why $OP\geq3\sqrt2$ for all $t>0$, and find the value of $t$ for which $OP=3\sqrt2$.

   [6 marks]

   Total for question 4: 12 marks

End of questions