Tangents and normals to the parabola and rectangular hyperbolaEdexcel International A Level Further Maths: Subtopic test

1. 1
   The parabola $P$ has equation $y^2=12x$.

   (a)

   Find the gradient of the tangent to $P$ at the point $(12,12)$.

   [1 mark]

   • A$2$
   • B$1$
   • C$\frac12$
   • D$6$

   (b)

   Find the gradient of the normal to $P$ at the point $(12,12)$.

   [1 mark]

   • A$2$
   • B$-\frac12$
   • C$\frac12$
   • D$-2$

   (c)

   Find the equation of the tangent to $P$ at the point $(12,12)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The rectangular hyperbola $H$ has equation $xy=24$ and passes through the point $A(4,6)$.

   (a)

   Find the gradient of the tangent to $H$ at $A$.

   [1 mark]

   • A$-\frac32$
   • B$\frac32$
   • C$-\frac23$
   • D$-6$

   (b)

   Find the gradient of the normal to $H$ at $A$.

   [1 mark]

   • A$-\frac23$
   • B$\frac23$
   • C$\frac32$
   • D$-\frac32$

   (c)

   Find the equation of the normal to $H$ at $A$, in the form $ax+by+c=0$ where $a$, $b$ and $c$ are integers.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The parabola $C$ has equation $y^2=8x$. The point $P$ lies on $C$, has $x$-coordinate 8 and has positive $y$-coordinate.

   (a)

   Find the equation of the tangent to $C$ at $P$.

   [3 marks]

   (b)

   The tangent at $P$ meets the $x$-axis at $Q$, and the normal at $P$ meets the $x$-axis at $R$. Find the area of triangle $PQR$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The rectangular hyperbola $H$ has equation $xy=16$. The points $P(2,8)$ and $Q(8,2)$ lie on $H$.

   (a)

   Find the equations of the tangents to $H$ at $P$ and at $Q$, and find the coordinates of the point where these two tangents meet.

   [6 marks]

   (b)

   The normal to $H$ at $P$ meets $H$ again at the point $R$. Find the coordinates of $R$.

   [6 marks]

   Total for question 4: 12 marks

End of questions