Intersections of

1
Chapter 8 Coordinate Geometry and Trigonometry
8.5
Intersections of Lines and Circles
8.5.1
MATHPOWERTM 11, WESTERN EDITION
2
Intersections of Lines and Circles - Solving
Algebraically
Find the coordinates of the points of
intersection of the line y x 1 and the
circle x2 y2 41.
Solve the system of equations y x 1
(1) x2 y2 41 (2)
Substitute x 1 for y in (2)
x2 (x 1)2 41 x2 x2 2x 1 41
2x2 2x - 40 0 x2 x - 20 0
(x 5)(x - 4) 0 x 5 0 or x - 4 0
x -5 x 4
Solving for y in y x 1
x -5 y -5 1 y -4
x 4 y 4 1 y 5
The points of intersection are (-5, -4) and (4,
5).
8.5.2
3
Intersections of Lines and Circles - Solving
Graphically
Find the coordinates of the points of
intersection of the line y x 1 and the
circle x2 y2 41.
From the graph, the points of intersection
are (4, 5) and (-5, -4).
8.5.3
4
Points of Tangency
For what values of c does the line y c touch
the circle x2 y2 r2 ?
y c
x2 y2 r2
The line y c is tangent to the circle x2 y2
r2 when c r.
8.5.4
5
Finding the Equation of the Tangent to the Circle
Find the equation of the tangent to the circle
(x - 2)2 (y 1)2 26 at the point A(3, 4).
Therefore, the slope of the tangent will be
A(3, 4)
Use the slope and the point A(3, 4)
C(2, - 1)
y - y1 m(x - x1) y - 4 (x - 3)
Find the slope of CA.
5y - 20 -1(x - 3) 5y - 20 -x
3 x 5y - 23 0
The equation of the tangent is x 5y - 23 0.
8.5.5
6
The Length of the Tangent
Find the length of a tangent from the point P(5,
10) to the circle x2 y2 25.
Find the length CP.
P(5, 10)
C
5
A
Using the Pythagorean Theorem, solve for AP
(CP)2 (AP)2 (CA)2 (5v5)2 (AP)2 52
125 (AP)2 25 100 (AP)2 10 AP
Therefore, the length of the tangent to the
circle is 10 units.
8.5.6
7
Assignment
Suggested Questions Pages 489 and 490 3, 8, 10,
20, 21, 24, 29, 34, 41, 47, 56, 62, 77, 83
8.5.7