Distance, Circles, And Quadratic Equations Worksheets Page 8
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9H8
Appendix H: Distance, Circles, and Quadratic Equations
EXERCISE SET H
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23–24
1. Where in this section did we use the fact that the same scale
Find the center and radius of each circle.
was used on both coordinate axes?
+ y
= 25
2
2
23. (a) x
(b) (x − 1)
+ (y − 4)
= 16
2
2
2–5
Find (a) the distance between A and B and (b) the mid-
(c) (x + 1)
+ (y + 3)
= 5
2
2
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point of the line segment joining A and B.
+ (y + 2)
= 1
2
2
(d) x
2. A(2, 5), B(−1, 1)
3. A(7, 1), B(1, 9)
+ y
= 9
2
2
24. (a) x
5. A(−2, −6), B(−7, −4)
4. A(2, 0), B(−3, 6)
(b) (x − 3)
+ (y − 5)
= 36
2
2
(c) (x + 4)
+ (y + 1)
= 8
2
2
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6–10
Use the distance formula to solve the given problem.
(d) (x + 1)
+ y
= 1
2
2
6. Prove that (1, 1), (−2, −8), and (4, 10) lie on a straight line.
7. Prove that the triangle with vertices (5, −2), (6, 5), (2, 2)
25–32
Find the standard equation of the circle satisfying the
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is isosceles.
given conditions.
8. Prove that (1, 3), (4, 2), and (−2, −6) are vertices of a right
25. Center (3, −2); radius = 4.
√
triangle and then specify the vertex at which the right angle
26. Center (1, 0); diameter =
8.
occurs.
27. Center (−4, 8); circle is tangent to the x-axis.
9. Prove that (0, −2), (−4, 8), and (3, 1) lie on a circle with
28. Center (5, 8); circle is tangent to the y-axis.
center (−2, 3).
29. Center (−3, −4); circle passes through the origin.
10. Prove that for all values of t the point (t, 2t − 6) is equidis-
30. Center (4, −5); circle passes through (1, 3).
tant from (0, 4) and (8, 0).
31. A diameter has endpoints (2, 0) and (0, 2).
11. Find k, given that (2, k) is equidistant from (3, 7) and (9, 1).
32. A diameter has endpoints (6, 1) and (−2, 3).
12. Find x and y if (4, −5) is the midpoint of the line segment
joining (−3, 2) and (x, y).
33–44
Determine whether the equation represents a circle, a
point, or no graph. If the equation represents a circle, find the
I
13–14
Find an equation of the given line.
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center and radius.
13. The line is the perpendicular bisector of the line segment
+ y
− 2x − 4y − 11 = 0
2
2
33. x
joining (2, 8) and (−4, 6).
+ y
+ 8x + 8 = 0
2
2
34. x
14. The line is the perpendicular bisector of the line segment
+ 2y
+ 4x − 4y = 0
2
2
35. 2x
joining (5, −1) and (4, 8).
+ 6y
− 6x + 6y = 3
2
2
36. 6x
15. Find the point on the line 4x − 2y + 3 = 0 that is equidis-
+ y
+ 2x + 2y + 2 = 0
2
2
tant from (3, 3) and (7, −3). [Hint: First find an equation of
37. x
+ y
− 4x − 6y + 13 = 0
2
2
the line that is the perpendicular bisector of the line segment
38. x
joining (3, 3) and (7, −3).]
+ 9y
= 1
2
/ 4) + (y
2
/ 4) = 1
2
2
39. 9x
40. (x
16. Find the distance from the point (3, −2) to the line
+ y
+ 10y + 26 = 0
2
2
41. x
(a) y = 4
(b) x = −1.
+ y
− 10x − 2y + 29 = 0
2
2
42. x
17. Find the distance from (2, 1) to the line 4x − 3y + 10 = 0.
+ 16y
+ 40x + 16y − 7 = 0
2
2
43. 16x
[Hint: Find the foot of the perpendicular dropped from the
+ 4y
− 16x − 24y = 9
2
2
44. 4x
point to the line.]
45. Find an equation of
18. Find the distance from (8, 4) to the line 5x + 12y − 36 = 0.
+ y
= 16
2
2
(a) the bottom half of the circle x
[Hint: See the hint in Exercise 17.]
+ y
+ 2x − 4y + 1 = 0.
2
2
(b) the top half of the circle x
19. Use the method described in Exercise 17 to prove that the
46. Find an equation of
) to the line Ax + By + C = 0 is
distance d from (x
, y
0
0
+ y
= 9
2
2
(a) the right half of the circle x
|Ax
+ By
+ C|
+ y
− 4x + 3 = 0.
2
2
(b) the left half of the circle x
0
0
d =
+ B
47. Graph
2
2
√
√
A
(a) y =
25 − x
(b) y =
5 + 4x − x
2
2
.
20. Use the formula in Exercise 19 to solve Exercise 17.
48. Graph
21. Use the formula in Exercise 19 to solve Exercise 18.
(a) x = − 4 − y
(b) x = 3 +
4 − y
2
2
.
22. Prove: For any triangle, the perpendicular bisectors of the
49. Find an equation of the line that is tangent to the circle
sides meet at a point. [Hint: Position the triangle with one
+ y
= 25
2
2
x
vertex on the y-axis and the opposite side on the x-axis, so
at the point (3, 4) on the circle.
that the vertices are (0, a), (b, 0), and (c, 0).]
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