1. Circles, Line intersections and Tangents © Christine Crisp Objective : To find intersection points with straight lines To know if a line crosses a circle To remind you of tangents Keywords Tangent, Discriminant, Distinct roots

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

3. Circles, Lines and Tangents To find the points of intersection of a line and circle we need to solve the equations simultaneously.

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

5. Circles, Lines and Tangents The quadratic equation will have no solutions if the line and circle don’t meet If the line does not cut the circle, there are no points of intersection. 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.

6. Circles, Lines and Tangents 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 3 rd 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.

7. Circles, Lines and Tangents 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.

8. Circles, Lines and Tangents 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)

9. Circles, Lines and Tangents Solution: (b) and Substitute in the linear equation: Exercise