Coordinate geometry practice questions (A-Level Maths)

Sample questions

The one free A-Level Maths question on this topic, from the free A-Level Maths Practice Paper. Each answer is behind its own toggle; the whole paper, its PDF and the full answer key are on the exam page.

1. Question 16 marks · Coordinate geometry: circles
   The circle $C$ has centre $(3, -2)$ and passes through the point $P(7, 1)$.
   (a)

   Find an equation for $C$.

   [3]
   (b)

   Find an equation of the tangent to $C$ at the point $P$, giving your answer in the form $ax + by + c = 0$, where $a$, $b$ and $c$ are integers.

   [3]
   Show answerFrom the free A-Level Maths Practice Paper →

Key terms

The definitions and formulas A-Level Maths expects for coordinate geometry, typeset as they appear on the paper.

Gradient and equation of a straight line

The gradient is the change in $y$ over the change in $x$. Given a point and a gradient, the point-gradient form is quickest; rearrange to $ax + by + c = 0$ if asked.

$$m = \frac{y_2 - y_1}{x_2 - x_1}, \qquad y - y_1 = m(x - x_1)$$

Parallel and perpendicular lines

Parallel lines have equal gradients; perpendicular lines have gradients whose product is $-1$. The perpendicular bisector of a segment passes through its midpoint.

$$m_1 m_2 = -1$$

Equation of a circle

A circle with centre $(a, b)$ and radius $r$. Complete the square in $x$ and in $y$ to read the centre and radius from the expanded form.

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

Circle theorems used in coordinate geometry

The tangent at a point is perpendicular to the radius to that point; the perpendicular bisector of a chord passes through the centre; the angle in a semicircle is a right angle, so a triangle with a diameter as hypotenuse has its vertex on the circle.

Distance between two points and midpoint

Pythagoras for the distance; averaging coordinates for the midpoint. Both appear in circle questions to find the centre and radius from a diameter.

$$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}, \qquad M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

Parametric equations

A curve given by $x = f(t)$ and $y = g(t)$. Eliminate $t$ (often with a trig identity) to find the Cartesian equation, and state the domain the parameter range produces.

$$x = f(t), \quad y = g(t), \qquad \frac{dy}{dx} = \frac{dy/dt}{dx/dt}$$

Frequently asked questions

Other A-Level Maths topics

• Proof, algebra and functions
• Sequences and series
• Trigonometry
• Exponentials and logarithms
• Differentiation
• Integration
• Numerical methods and vectors
• Sampling and data presentation
• Probability and distributions
• Hypothesis testing
• Kinematics
• Forces, Newton's laws and moments

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