Circular Connections
Name_______________________
Period_____Date______________
1. If A has coordinates (4, -2) and B has coordinates (-6, -14), write the equation of the
circle that has diameter AB.
(x + 1)2 + (y + 8)2 = 61
2. Write the equation of the circle that is tangent to both the positive x-axis and the
positive y-axis if the area of this circle is 49π? (x – 7)2 + (y – 7)2 = 49
3. Write the equation of the circle that is tangent to both the positive x-axis and the
positive y-axis if the area of this circle is 24π?

x 2 6
 
2
 y 2 6

2
 24
4. Write the equation of two circles with circumference 16π that are tangent to the xaxis at (5, 0). r = 8, so (x – 5)2 + (y – 8)2 = 64 & (x – 5)2 + (y + 8)2 = 64
5. If A has equation (x – 6)2 + (y + 2)2 = 64, and a chord of A is constructed that is 5
units away from A, how long is that chord? Radius is 8, so (c/2)2 + 52 = 64. c = 2 39
6. Find the center and radius of x2 + y2 – 8x + 4y = 80.
(x – 4)2 + (y + 2)2 = 100, so
center is (4, -2) and radius is 10.
7. Find the center and radius of x2 + y2 + 14x – 9y = 11.75.
(x + 7)2 + (y – 4.5)2 = 81,
so center is (-7, 4.5) and radius is 9.
8. Find the center and radius of 9x2 + 9y2 = 64. Center is (0, 0) and radius is 8/3.
9. Find the center and radius of 4x2 + 4y2 + 8x – 24x = 9 (x + 1)2 + (y – 3)2 = 12.25
so the center is (-1, 3) and the radius is 3.5 .
10. Write the equation of the line that is tangent to (x – 3)2 + (y + 2)2 = 169 at (-2, 10).
Slope from (3, -2) to (-2, 10) is 12/-5, so m() = 5/12.
y = 5x/12 + 65/6
11. Write the equation of the circle with center (4, 0) that is tangent to both y = ½ x + 3
and y = - ½ x – 3. m() = -2, so radii are on y = -2x + 8. Solve ½ x + 3 = -2x + 8 to
find point of tangency is (2, 4). So radius2 is 20  (x – 4)2 + y2 = 20.
12. Circles P and Q are tangent to each other
and to the axes as shown. If PQ = 26 and AB = 24,
P
find the coordinates of P and Q. ∆PQM is a rt.∆
M
so PM = 10. If radius of small = x, then
A
Q
B
PQ = 2x + 10 = 26. Therefore x = 8 and radius of large  is 18. So P = (18, 18)
and Q = (42, 10).
13. A circle is inscribed in a triangle A whose vertices are (0, 0), (24, 0) and (0, 18).
Write an equation of this circle.
AF = 18 – r = AE. Also CG = CE =
E
F
B
D
G
BGDF is a square of radius r. So
C
24 – r. This means AC =
(24 – r) + (18 – r) = 30, so r = 6, so equation is (x – 6)2 + (y – 6)2 = 36.
14. A regular hexagon is inscribed in the circle determined by x2 + y2 + 4x = 32. Find the
area of this hexagon. (x + 2)2 + y2 = 36, so r = 6. The hexagon is 6
equilateral triangles, so A = 6(½ 6 3 3 ) = 54 3
15. Write an equation of any circle that is tangent to x2 + y2 + 12y = 45 and explain how
you know the circles are tangent to each other. x2 + (y + 6)2 = 81, so center is (0, -6)
and r = 9.
(0, 3) lies on this circle, so a circle with radius 2 with center 2 units
above (0, 3) will only intersect this circle in one point. x2 + (y – 5)2 = 4 will work.A
16. Segments AB and CD are both chords of circle K. If A = (8, 13), B = (8, -3),
C
C = (-6, 11) and D = (10, 11), write the equation of K. Since AB ‖ y axis and
D
K
CD ‖ x axis, K has coordinates (2, 5). KA2 = 62 + 82 = 100. (x – 2)2 + (y – 5)2 = B100
17. If M = (-4, -2), N = (14, 10) and P = (22, -2), and if MNP is inscribed in a semicircle
with center Q, MP must be a diameter, so center is (9, -2) and radius is 13.
A) Write the equation of Q.
(x – 9)2 + (y + 2)2 = 169
B) Find the area of ΔMNP. A = ½ (26)(12) = 156.
C) Why is MNP a right angle? The measure of an inscribed angle is half the
measure of the intercepted arc and a semicircle measures 180˚. AND m(MN) = 2/3
while m(NP) = -3/2.
18. Consider the circle defined by the equation x2 + y2 – 14x = 1 and the line l that has
equation y = x – 7. Find the points of intersection between l and C. x2 + (x - 7)2 – 14x
= 1  x = 12 or 2, so points are (12, 5) and (2, -9).
19. What is the shortest distance between (x - 2)2 + (y – 2)2 = 4 and the origin? 2 2  2
20. What is the shortest distance between the graph of x2 + y2 - 16x + 6y + 24 = 0 and the
point (-5, 5)? (x – 8)2 + (y + 3)2 = 49. The distance between (8, -3) and (-5, 5) is
233 so the shortest distance is
233 - 7.
21. Write the equation for the locus of points that is 8 units from (7, -12).
(x – 7)2 + (y + 12)2 = 64
22. If A and B lie on x2 + y2 + 4x – 2y = 31, and if the measure of arc AB = 120˚,
A) Find the length of arc AB. (x + 2)2 + (y – 1)2 = 36, so r = 6. The length of
arc AB = 120/360(2πr) = 1/3(12π) = 4π units
B) Find the area of the sector determined by arc AB. The area of the sector is
1/3(πr2) = 1/3π36 = 12π square units.