Aim #17: How do we write equations of lines tangent to circles?
2
CC Geometry H
2
Do Now: Consider the circle (x - 1) + (y - 2) = 16. There are two lines tangent
to this circle having a slope of 0.
a) Find the coordinates of the two points
of tangency.
b) Find the equation of the two tangent lines.
Recall:
THEOREM: A tangent line to a circle is perpendicular to the
radius of the circle drawn to the point of tangency
A tangent line to a circle is a line in the same plane that intersects the circle
at one and only one point. This point is called the point of tangency.
1. a) What are the coordinates of the points of tangency of the two tangent lines
2
2
through the point (1,1) each tangent to the circle x + y = 1?
b) What are the coordinates of the points of
tangency of the two tangent lines through
the point (-1,-1) each tangent to the circle
2
2
x + y = 1?
2. a) Write the equation, in point-slope form, of the line tangent at the point
2
2
(5,7) to the circle with equation (x - 2) + (y - 3) = 25.
b) Two tangent segments drawn to the
circle from an external point intersect the
circle at (-3,3) and (2,-2). Namethe
coordinates of the external point and find
the lengths of the tangent segments.
There should be two lines tangent to a circle for any given slope.
Example: Consider the circle with equation, (x - 3)
2
2
+ (y - 5) = 20. Find the
equations of two tangent lines to the circle that each have a slope of -½.
a) What is the center of the circle and the radius? ______________________
b) If tangent lines have a slope of -½, what must be the slope of the radii to those
tangent lines?
c) Using the center point, write a slope equation. We need to find the points on
the circle which have a slope of 2. Solve for the points of tangency by first
finding the x-values.
d) The coordinates of the points of tangency: _________ _________
e) Verify that these coordinates lie on the circle.
e) Write the equations of the 2 tangent lines which have a slope of -½.
2
2
3. Consider the circle (x - 5) + y = 10. There are two lines tangent to this circle
having a slope of .
a) Find the coordinates of the two points of tangency.
b) Find the equation of the two tangent lines.
2
2
4. Show that the circle with equation (x- 2) + (y + 3) = 160 has two tangent lines
with equations y + 15 = (x - 6) and y - 9 = (x + 2) .
(­2,9)
(2,­3)
(6,­15)
Name___________________________
Date: ________________
1) Consider the circle with equation, (x - 4)
CC Geometry H
HW #17
2
2
+ (y - 5) = 20. There are two lines
tangent to this circle having a slope of 2.
a) Find the coordinates of the two points of tangency.
b) Find the equation of the two tangent lines.
2) What are the coordinates of the points of tangency of the two tangent lines
2
2
through the point (5,5) each tangent to a circle x + y = 25.
3) What are the coordinates of the points of tangency of the two tangent lines
2
2
through the point (-5, -5) each tangent to the circle x + y = 25 ?
4) Write an equation of a circle whose center is (−3, 2) and whose diameter is 10.
5) The coordinates of the endpoints of the diameter of a circle are (2, 0) and
(2, −8). What is the equation of the circle?
Mixed Review:
1) Find the exact volume of this figure formed by a cylinder and a cone.
2) Solve for θ:
0
a) cos 60 = sin θ
c) sin θ = cos(θ + 20)
0
b) sin 71 = cos θ
d) sin (θ - 60) = cosθ
3) Solve for x and y:
4) Which of the following is NOT a way to prove a quadrilateral is a parallelogram?
(1) Show both sets of opposite angles of the quadrilateral are congruent.
(2) Show the diagonals of the quadrilateral bisect each other.
(3) Show one set of opposite sides of the quadrilateral is both congruent and
parallel.
(4) Show one set of opposite sides of the quadrilateral is congruent.