Pure Mathematics Practice: Conic Sections

Original practice material

These questions total 24 marks. They assess conics only as graphing-support knowledge within the 9758 scope; focus-directrix proofs and deeper conic geometry are not required.

Answers: Conic Sections Answers.

Question 1: completing the square and a circle

The curve $C$ has equation

$$x^2+y^2-6x+4y-12=0.$$

(a) Express the equation of $C$ in the form

$$(x-a{)}^2+(y-b{)}^2=r^2.$$

Hence state the centre and radius of $C$. [3]

(b) Find the exact coordinates of the points where $C$ meets the $x$-axis. [2]

(c) State the range of possible values of $y$ on $C$. [1]

Question 2: ellipse structure and sketch

The curve $E$ has equation

$$\frac{(x+1{)}^2}{16}+\frac{(y-2{)}^2}{9}=1.$$

(a) State the centre of $E$ and the lengths of its horizontal and vertical semi-axes. [2]

(b) State the possible values of $x$ and of $y$. [2]

(c) Sketch $E$, showing its centre and its four vertices. [2]

Question 3: hyperbola structure and intersections

The curve $H$ has equation

$$\frac{(y+1{)}^2}{4}-\frac{(x-2{)}^2}{9}=1.$$

(a) State the centre, the direction in which the branches open, and the coordinates of the vertices. [3]

(b) Write down the equations of the two asymptotes. [2]

(c) Find the exact coordinates of the points where $H$ meets the $y$-axis. [3]

(d) Sketch $H$, showing its centre, vertices and asymptotes. [2]

Question 4: parabola from completed-square form

The equation

$$y^2-4y-8x+12=0$$

represents a parabola.

(a) Show that its equation may be written as

$$(y-2{)}^2=8(x-1).$$

[1]

(b) State the vertex, axis of symmetry and the possible values of $x$. [2]

(c) Sketch the parabola, showing the vertex and its intercept with the $x$-axis. [1]