17: Circles, Lines and Tangents “Teach A Level Maths” Vol. 1: AS Core Modules 17: Circles, Lines and Tangents © Christine Crisp

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The graph shows that the line and circle meet at the points (-2, -1) and (0, 1). If a line cuts a circle, the coordinates of the points of intersection satisfy the equations of the line and the circle e.g. Substituting the coordinates of the point (-2, -1) into the equations: Both equations are satisfied by (-2, -1)

To find the points of intersection of a line and circle we need to solve the equations simultaneously.

e.g. Find the coordinates of the points where the line cuts the circle Solution: Substitute for y from the linear equation into the quadratic equation: This is a quadratic equation so we need to simplify and get 0 on one side, then try to factorise Taking out the common factors: Notice that the discriminant, of this quadratic equation equals Since 16 is > 0, the equation has real, distinct roots. Substituting in the linear equation: and

If the line does not cut the circle, there are no points of intersection. The quadratic equation will have no solutions if the line and circle don’t meet e.g. Consider the line and circle The discriminant, Since , the equation has no real roots If we try to solve the equation, we get which also shows there are no real solutions.

The discriminant of the quadratic equation has shown us whether the line cuts the circle in 2 places or does not meet the circle. The 3rd possibility is that the line just touches the circle. It is then a tangent. In this case the discriminant equals 0 and the quadratic equation has equal roots.

SUMMARY The discriminant of the quadratic equation formed by eliminating y from the equations of a straight line and a circle tells us how the line and circle are related. 2 points of intersection: No points of intersection: Tangent:

Exercise Use the discriminant of a quadratic equation to determine whether the following lines meet the circle If so, find the points of intersection (a) (b) Solution: (a)

Exercise (b) and Solution: Substitute in the linear equation:

The following slide contains repeats of information on earlier slides, shown without colour, so that it can be printed and photocopied.

Tangent: No points of intersection: 2 points of intersection: SUMMARY The discriminant of the quadratic equation formed by eliminating y from the equations of a straight line and a circle tells us how the line and circle are related.