Algebra - 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

This is Higher tier only. A circle of radius r drawn about the origin has equation x² + y² = r², which is Pythagoras’ theorem a² + b² = c² applied to the right-angled triangle under any point on the circle. So x² + y² = 25 is the circle of radius 5, and the point (3, 4) lies on it because 9 + 16 = 25. A tangent touches the circle at one point and is at right angles to the radius drawn to that point. To find its equation, work out the gradient of that radius from the origin, take the negative reciprocal for the tangent, then substitute the point. If the point of contact sits on an axis, the radius is horizontal or vertical and the tangent there is the vertical or horizontal line through the point. At (3, 4) the radius has gradient 4/3, so the tangent has gradient −3/4, and 4 = −9/4 + c gives c = 25/4, making the tangent y = −3x/4 + 25/4.

What earns the marks

• The radius found from the equation, or for the equation written from a given radius.
• Showing a point lies on the circle by substituting both of its coordinates.
• The gradient of the radius drawn to the point of contact.
• The negative reciprocal taken as the gradient of the tangent.
• The finished tangent equation, in whichever form the question asks for.

Common mistakes and how to put them right

• Giving the radius as 25 rather than 5 for the circle x² + y² = 25.

• Using the gradient of the radius as the gradient of the tangent.

• Turning the gradient upside down without changing its sign, or changing the sign without turning it upside down.

• Assuming a centre somewhere other than the origin, when this content covers only circles about the origin.

Exam tips

• Take the radius as the square root of the number on the right of the equation, so x² + y² = 49 has radius 7.
• Sketch the circle and mark the point of contact, and the right angle between radius and tangent becomes obvious.
• Keep fractions as fractions all the way through, because a rounded gradient makes the final equation wrong.