Circles, Line Circles, Line intersections and intersections and

TangentsTangents

© Christine Crisp

Objective :

• To find intersection points with straight lines• To know if a line crosses a circle• To remind you of tangents

KeywordsTangent,

Discriminant, Distinct roots

Circles, Lines and Tangents

1xy

4)1()2( 22 yx

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).

1xy 4)1()2( 22 yx

e.g. Substituting the coordinates of the point (2, 1) into the equations:

l.h.s.r.h.s. gives in 1121)1,2( xy

4)1()2( 22 yxin )1,2( 22 )11()22( l.h.s. gives

4 r.h.s.Both equations are satisfied by (-2, -1)

Circles, Lines and Tangents

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

Circles, Lines and Tangents

0)2(2 xx

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.

acb 42 16)0)(2(416

4)1()2(1 22 xxy 1x

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 circle1xy 4)1()2( 22 yx

This is a quadratic equation so we need to simplify and get 0 on one side, then try to factorise

4)2)(2( 2 xxx0444 22 xxx042 2 xx

20 xx or

0x 1y 2x 1yand

4)1()2( 22 yx

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.

4)1()2(1 22 xxy 1x4)2)(2()2)(2( xxxx44444 22 xxxx042 2 x

e.g. Consider the line and circle 4)1()2( 22 yx1xy

The discriminant, 032)4)(2(4042 acb

Since , the equation has no real roots 042 acb

If we try to solve the equation, we getwhich also shows there are no real solutions.

22 x

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 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.

Circles, Lines and Tangents

Tangent:

042 acb

No points of intersection:

042 acb

2 points of intersection: 042 acb

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.

Circles, Lines and TangentsExercis

eUse the discriminant of a quadratic equation to determine whether the following lines meet the circle

If so, find the points of intersection

04222 yxyx

(a) (b) xy 2 92 xy

Solution: (a)

0)2(42)2( 22 xxxx

0824 22 xxxx

0105 2 xx

0)2(5 xx

100)0)(5(4)10(4 22 acb

onintersecti of points 2 042 acb

20 xx or

4200 yxyx and

Circles, Lines and Tangents

Solution: 0)92(42)92( 22 xxxx

03682)92)(92(2 xxxxx

0)96(5 2 xx

0900900)45)(5(4)30(4 22 acb

circle the totangent a is line the 042 acb

3 x33 yx

0368281364 22 xxxxx045305 2 xx

0)3)(3(5 xx

04222 yxyx(b) 92 xyand

Substitute in the linear equation:

Exercise