Graphs: recognise and use the equation of a circle with centre at the origin find the equation of a tangent to a circle at a given point (Higher tier only)

The idea

A circle with centre at the origin and radius r has equation x² + y² = r², so a point lies on the circle exactly when its coordinates satisfy that equation. To recognise the equation, compare the constant term with the square of the radius; for example x² + y² = 25 is a circle of radius 5 centred at the origin. To find the equation of a tangent at a given point, first find the gradient of the radius joining the origin to that point, then use the fact that the tangent is perpendicular to the radius, so its gradient is the negative reciprocal. Substitute the point's coordinates into y = mx + c, or use the point-gradient form, to obtain the tangent's equation.

What earns the marks

• Recognises that x² + y² = r² represents a circle with centre at the origin and radius r.
• Uses the equation to test whether a given point lies on, inside or outside the circle.
• Finds the gradient of the radius from the origin to the given point on the circle.
• Uses perpendicularity to obtain the tangent gradient as the negative reciprocal of the radius gradient.
• Forms the equation of the tangent through the given point, for example by substituting into y = mx + c or using the point-gradient form.
• Handles special cases such as a point on an axis, where the radius is vertical or horizontal and the tangent is the other orientation.

Common mistakes and how to put them right

• Reading the constant term as the radius rather than the radius squared.

  Put it right: x² + y² = 25 has radius 5, since 5² = 25.

• Using the radius gradient itself as the tangent gradient.

  Put it right: The tangent is perpendicular to the radius, so its gradient is the negative reciprocal.

• Forgetting the negative sign when taking the negative reciprocal, for example turning a radius gradient of 2 into a tangent gradient of 2 rather than −1/2.

• Substituting the origin instead of the given point when finding the tangent's intercept.

  Put it right: The tangent passes through the given point on the circle.

Exam tips

• Write down the radius gradient and the tangent gradient as separate steps before forming the tangent equation, so the perpendicular relationship is clearly shown.
• Check your tangent equation by substituting the given point's coordinates; both sides should agree.
• For a point on an axis, sketch the circle and tangent first, since the tangent is then a simple horizontal or vertical line.