Circles, Line intersections
and Tangents
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
© Christine Crisp
Circles, Lines and Tangents
The graph shows that the line
y  x  1 and circle ( x  2) 2  ( y  1) 2  4
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
2
2
( x  2)  ( y  1)  4
y  x 1
e.g. Substituting the coordinates of the point (2, 1)
into the equations:
(2,1) in
y  x  1 gives r.h.s.  2  1  1  l.h.s.
( 2,1) in ( x  2)  ( y  1)  4
2
2
gives l.h.s.  ( 2  2)  ( 1  1)
2
 4  r.h.s.
2
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
e.g. Find the coordinates of the points where the
line y  x  1 cuts the circle ( x  2) 2  ( y  1) 2  4
Solution: Substitute for y from the linear equation into
the quadratic equation:
2
( x  2)  ( y  1)
y  x 1

2
4
2
2
( x  2)  ( x  1  1)  4
2
( x  2)( x
2
)  need
x  4to simplify

This is a quadratic
equation
so
we
2
2
 side,
x then
 4 x try
4 tox factorise
4  0
and get 0 on one
2

2x  4x  0
Taking
out that
the the
common
factors: b 2  4ac of this
Notice
discriminant,

2 x( x
2)0
quadratic equation
equals 16
4(2)(
) 0 16
x
0
or has
x real,
 2 distinct roots.
Since 16 is >0, the
equation
Substituting in the linear equation:
and x  2  y  1
x 0  y 1
Circles, Lines and Tangents
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 y  x  1 and circle ( x  2) 2  ( y  1) 2  4
y  x 1
2
2
( x  2)  ( x  1  1)  4


( x  2)( x  2)  ( x  2)( x  2)  4

x  4x  4  x  4x  4  4

2x  4  0
2
2
2
The discriminant, b 2  4ac  0  4(2)( 4)  32  0
Since b 2  4ac  0 , the equation has no real roots
If we try to solve the equation, we get x 2  2
which also shows there are no real solutions.
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
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:
2
No points of
b  4ac  0
intersection:
2
b  4ac  0
Tangent: b 2  4ac  0
Circles, Lines and Tangents
Exercise
Use the discriminant of a quadratic equation to
determine whether the following lines meet the circle
2
2
x  y  2x  4 y  0
If so, find the points of intersection
(b) y  2 x  9
(a) y  2 x
Solution: (a) x 2  ( 2 x ) 2  2 x  4(2 x )  0
2
2
x  4x  2x  8x  0
2
5 x  10 x  0
2
2
b  4ac  ( 10)  4(5)( 0)  100
b  4ac  0  2 points of intersecti on
2
5 x( x  2)  0
 x0
x0 
y0
and
or
x2
x2  y4
Circles, Lines and Tangents
Exercise
(b)
2
2
x  y  2 x  4 y  0 and y  2 x  9
Solution: x 2  (2 x  9) 2  2 x  4(2 x  9)  0
2
x  ( 2 x  9)( 2 x  9)  2 x  8 x  36  0
2
2
x  4 x  36 x  81  2 x  8 x  36  0
2
5 x  30 x  45  0
2
2
b  4ac  ( 30)  4(5)( 45)  900  900  0
b  4ac  0  the line is a tangent to the circle
2
2
5( x  6 x  9)  0
5( x  3)( x  3)  0
 x3
Substitute in the linear equation: x  3  y  3