Circles in General Form

Show your work for each problem.

1.
Which equation is in general form?
1. a)
2. b)
3. c)
4. d)
Answer:
is in general form with , , . The others are in standard form.
2.
For the equation , what is the value of ?
Answer: 6
In general form , is the coefficient of . Here, the term is , so .
3.
What is the center of a circle with equation ?
1. a)
2. b)
3. c)
4. d)
Answer:
With and : and . Center: .
4.
For the equation , what is ?
Answer: -2
The term is , so . Therefore . This is the -coordinate of the center.
5.
Convert to standard form.
Answer: (x-1)^2 + (y+3)^2 = 16
1. What value do you add to complete the square for ? 1
2. What value do you add to complete the square for ? 9
3. What is the right side after adding to both sides? (Starting from 6) 16
4. What is the -coordinate of the center? 1
6.
Find the radius of the circle .
Answer: 5
.
7.
What is the standard form of ?
1. a)
2. b)
3. c)
4. d)
Answer:
. So .
8.
Find the center and radius of .
Answer: center: (-6, 2), radius: 7
1. What is where ? -6
2. What is where ? 2
3. Calculate 36
4. Calculate 4
5. Calculate 7
9.
A circle has center and radius . What is the value of in general form?
Answer: -6
expands to , which gives . So .
10.
Does represent a real circle?
1. a)No, it represents no real circle
2. b)No, it represents a point
3. c)Yes, with radius 5
4. d)Yes, with radius 3
Answer: No, it represents no real circle
. Since this is negative, no real circle exists.
11.
Convert to standard form and determine if the point is inside, on, or outside the circle.
Answer: inside
1. What is the center of the circle? (format: x, y) 4, -1
2. What is ? 9
3. What is the radius? 3
4. What is the distance from center to point ? sqrt(2)
5. Is less than, equal to, or greater than 3? less than
12.
Two circles have equations and . What is the difference in their radii?
Answer: 1
Both have . Circle 1: . Circle 2: . Difference: .
13.
What is the equation of a circle passing through the origin with center ?
1. a)
2. b)
3. c)
4. d)
Answer:
Radius . Standard form: . Expanding: , so .
14.
Find the equation in general form of a circle with center that passes through point .
Answer: x^2 + y^2 - 4x + 6y - 12 = 0
1. What is the distance from to ? 5
2. What is where ? -4
3. What is where ? 6
4. What is ? -12
15.
What does represent when we expand standard form to general form?
1. a)
2. b)
3. c)
4. d)
Answer:
expands to . The constant term is .
16.
The equation represents a circle. If , , and the circle passes through the origin, what is ?
Answer: 0
Substituting : , so . The equation becomes .