Name ________________________
Worksheet 2.5
Tangents and Intersections
Clearly, a line may intersect a circle zero, one, or two times.
A tangent to a circle is defined as a line which intersects the circle in exactly one point. The tangent is also perpendicular to the radius of the circle at the point of tangency and this fact enables us to easily define a tangent to a given circle at a given point. Write down the equation of the tangent to the circle x
2  y
2 
25 at the point (3,4).
Draw a picture of this scenario and also draw two other lines on the plane one of which intersects the circle in two points and the other does not intersect the circle at all. How does this picture relate to the three options for the solution of a quadratic equation?

To find if/where a line and a circle intersect, we must solve a system of equations.
Consider the following:
Q: Suppose we are given a circle ( x

1)
2 
( y

1)
2 
25 and the line y

4
3 x

6 . Analytically
(without graphing), determine how many times does this line intersect the circle?

Q: Suppose we are given the same circle ( x

1)
2 
( y

1)
2 
25 , but now the line 3 x y 3 .
Analytically (without graphing), determine how many times does this line intersect the circle?
By now you should have recognized to solve this system of equations requires working with the quadratic formula.
The value of the discriminant b
2 
4 ac determines how many solutions we have to our system of equations which subsequently tells us the number of points of intersection of a line and a circle.
Q: What should we anticipate to be true about the discriminant when a line does not intersect a circle?

Intersection of Circles: A Look at Orthogonality
Two different circles can have zero, one, or two points of intersection. Similar to the intersection of a line and a circle, if two circles meet at only one point then they are tangent to each other.
Exercise 1: Consider the following circles (named by their center-points):
Circle P: ( x

5)
2 
( y

5)
2 
25 AND Circle Q:
( x

5)
2 
( y

10)
2 
100 a.
Sketch these circles on the graph paper provided. b.
These two circles intersect at T passing through T. Draw this line on your diagram. c.
Analytically verify that the center point P lies on this tangent line to circle Q. d.
Find the equation of the line tangent to circle P passing through T. Draw this line on your diagram. e.
Analytically verify that the center point Q lies on this tangent line to circle P.

f.
Find the length of PQ . What relationship should hold between the lengths of PQ , PT , and QT ? Verify this relationship using coordinate techniques. g.
Comment on the validity of the following statement. Briefly explain your reasoning.
If two circles intersect at two points so that their respective tangent lines at a point of intersection are perpendicular, then their radii form right angles at this point of intersection.
We are now ready to define the following:

Orthogonal Circles: Two circles that meet at two points and have their radii form right angles at those points are known as orthogonal circles.
You might be (and perhaps should be) wondering if it is even possible to talk about orthogonal (a more mathematically sophisticated way of communicating perpendicular ) circles. After all, circles are “curves” and we have only discussed perpendicularity when working with lines, right?
Let’s explore this idea by taking a good look at the potential behavior of the circles at a point of intersection by graphing these circles on our calculators:
Task 1 : In your Y= Editor , type in the following to construct circle P (think about where these equations came from):
Y 1

25

( x

5)
2 
5
Y 2
 
25

( x

5)
2 
5
Task 2 : Also in your Y= Editor , type in the following to construct circle Q (you should now understand where these equations came from):
Y 3

100

( x

5)
2 
10
Y 4
 
100

( x

5)
2 
10
Task 3 : Press the ZOOM button and select option 5 (ZSquare). You should see your circles -- you may need to adjust your WINDOW to view the entire circles.
Task 4 : Move the cursor to an area close to the point circles we are studying.
Task 5 : Press the ZOOM button and select option 2 (Zoom In).
Task 6 : Press ENTER once your cursor is near the point of intersection
Task 7 : Repeat tasks 5 & 6 a couple more times…
Question
: How do the circles begin to “behave” at this point of intersection? Briefly comment on what you see and how this relates to the orthogonality of circles.

Let’s now look at exercises pertaining to orthogonality of circles:
Exercise 2: Given the following equations for two circles: x
2 
16 x
 y
2 
4 y

43

0 AND 2 x
2 
12 x

2 y
2 
40 y

70

0 a.
First, analytically find the center and radius of each circle. b.
Complete a quick sketch of these circles on the graph paper provided. c.
Without finding the points of intersection of these circles and only using the center point and radii of each circle, analytically determine if these two circles are orthogonal.

Finally, let’s go back to our initial set of intersecting circles and apply our coordinate geometry knowledge to explore a couple of other concepts related to circles:
Exercise 3: Consider the circles from exercise 1 (refer to your diagram):
Circle P: ( x

5)
2 
( y

5)
2 
25 AND
( x

5)
2 
( y

10)
2 
100
Circle Q: a.
We were given that the point T
Analytically find the other point of intersection of these circles (think about how you are going to do this using the two equations above – be patient with this process). b.
Verify that the circles are indeed orthogonal at the point of intersection you just found in part a .

c.
Find the equation of the common chord using the two points of intersection. This equation should look very familiar. Where have you seen this equation before? d.
Find the equation of PQ , the line connecting the center-points of the circles. e.
Analytically verify that PQ is the perpendicular bisector of the common chord.

1.
PXQ is tangent to circle C with coordinates as drawn. a.
Find the radius of circle C. b.
Find the slope of the tangent PXQ. c.
Find the equation of the tangent line PXQ (in point-slope form) d.
Find the equation for circle C. e.
Where does tangent PXQ intersect the x - and y - axes?
2.
C
P
(-2, 5) y
X (4, 9)
Q x
A circle equation has equation
3 x 2  12 x  3 y 2  6 y  2
. Show that the center of the circle has coordinates
  2,1

and radius
51
.
3

3. If the points (0,0), (5, 3), and (8, 0) are points on a circle find the coordinates of the center of the circle. Find the length of the radius. Try to solve this two ways: geometrically using Sketchpad, and then algebraically.

4. Given as shown in the diagram: Circle with center P: (3, 2), R: (9, 4) a point on the circle, V a point on the circle and on the x-axis, T: (15, -14); TR is tangent to the circle at R.
Find the length of the radius of circle P in exact terms. a.
Write an equation for circle P. y b.
Find the coordinates of point V.
V
P
(3, 2)
R (9, 4) x
T (15, -14) c.
Give the equation of tangent RT (use point-slope form) d.
Find the area of triangle PRT.

5. Circle Q is tangent to the x -axis and the y -axis at point B. Circle P is tangent to circle Q and the y -axis at point A. PQ = 26 and AB = 24. a) Find the coordinates of P and Q. b) Write the equations of both circles. y
24
A
26
P
B Q x

6. Write each of the following equations in center-radius form. Identify the center and the radius.
7. a. b. x
2  y
2 
4 x

2 y

7

0
4 x
2 
4 y
2 
16 x

4 y

17

0 c. 3 x
2 
3 y
2 
6 x

8 y

5

0
Identify the center and radius of the following circles. Write each of the following equations in standard form. a. b.
( x

4 )
2 
( y

3 )
2 
4
( x

1
  
2
2 y
2
) ( 2) 8

8. Finding intersections of circles and lines: a. Consider the circle with radius 5 which is tangent to both axes. Find the points on the circle where the diameter with slope -1 cuts the circumference of the circle. b. The circle ( x

4)
2 
( y

3)
2 
9 is tangent to the x-axis. Find the equation of the other line through the origin (besides the x-axis) which is tangent to the circle.

9.
Problems 1
Show that the circles with equations ( x  1) 2  ( y  3) 2  9 and ( x  2) 2  ( y  1) 2  16 are orthogonal.
10. You are given the circle with equation x 2  y 2  2 x  4 y  
and point A(5, 1). a.
Show that point A is on the circle. b.
Find the equation of the line tangent to the circle at point A.
1 This set of problems is from The Core Course for A-level

11. If the line with equation y 2 x c  
is tangent to the circle with equation x 2  y 2  4 x  10 y 7 0 , find the value(s) of c .
12. Given circles with equations x 2  ( y  1) 2  1 and ( x  1) 2  y 2  1 , find the equation of the common chord.