Tangents, normals and lociEdexcel International A Level Further Maths: Subtopic test

1. 1
   The ellipse $E$ has equation $\frac{x^2}{25}+\frac{y^2}{9}=1$ and $P$ is the point $\left(4,\frac95\right)$ on $E$.

   (a)

   Find an equation of the tangent to $E$ at $P$.

   [1 mark]

   • A$4x+3y=5$
   • B$4x-5y=25$
   • C$4x+5y=25$
   • D$25x-20y=64$

   (b)

   Find the gradient of the normal to $E$ at $P$.

   [1 mark]

   • A$\frac54$
   • B$-\frac45$
   • C$\frac45$
   • D$-\frac54$

   (c)

   The normal to $E$ at $P$ meets the $x$-axis at $A$. Find the $x$-coordinate of $A$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The hyperbola $H$ has equation $\frac{x^2}{16}-\frac{y^2}{9}=1$.

   (a)

   The line $y=2x+c$ is a tangent to $H$. Find the possible values of $c$.

   [1 mark]

   • A$\pm\sqrt{23}$
   • B$\pm2\sqrt5$
   • C$\pm\sqrt{73}$
   • D$\pm\sqrt{55}$

   (b)

   The point $\left(5,\frac94\right)$ lies on $H$. Find the gradient of the tangent to $H$ at this point.

   [1 mark]

   • A$\frac45$
   • B$\frac54$
   • C$-\frac54$
   • D$\frac{45}{16}$

   (c)

   Show that no tangent to $H$ has gradient $\frac12$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The ellipse $E$ has equation $\frac{x^2}{20}+\frac{y^2}{5}=1$ and $P$ is the point $(2,2)$ on $E$.

   (a)

   Find an equation of the normal to $E$ at $P$.

   [3 marks]

   (b)

   The normal to $E$ at $P$ meets $E$ again at $Q$. Find the coordinates of $Q$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A point $P(x,y)$ moves so that its distance from the point $S(3,0)$ is half its perpendicular distance from the line $x=12$.

   (a)

   Show that the locus of $P$ is an ellipse, and write its equation in the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.

   [6 marks]

   (b)

   The line $y=x+k$ is a tangent to this locus. Find the possible values of $k$ and hence the perpendicular distance between the two tangents with gradient $1$.

   [6 marks]

   Total for question 4: 12 marks

End of questions