12 multiple-choice questions, progressively harder.
What is the radius of the circle x2+y2−8x+2y+8=0x^2 + y^2 - 8x + 2y + 8 = 0x2+y2−8x+2y+8=0?
Solution
Correct answer: D
Complete the square: half of −8-8−8 is −4-4−4 (square 161616), half of 222 is 111 (square 111).
(x−4)2+(y+1)2=−8+16+1=9(x - 4)^2 + (y + 1)^2 = -8 + 16 + 1 = 9(x−4)2+(y+1)2=−8+16+1=9
The radius is 9=3\sqrt{9} = 39=3.
What is the distance between the points (−3,−2)(-3, -2)(−3,−2) and (5,4)(5, 4)(5,4)?
Correct answer: C
The horizontal gap is 5−(−3)=85 - (-3) = 85−(−3)=8 and the vertical gap is 4−(−2)=64 - (-2) = 64−(−2)=6.
d=82+62=64+36=100=10d = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10d=82+62=64+36=100=10
The distance is 101010.
A diameter of a circle has endpoints (1,2)(1, 2)(1,2) and (7,10)(7, 10)(7,10). What is the radius?
Correct answer: A
The radius is half the length of the diameter. First find the distance between the endpoints.
d=(7−1)2+(10−2)2=36+64=10d = \sqrt{(7 - 1)^2 + (10 - 2)^2} = \sqrt{36 + 64} = 10d=(7−1)2+(10−2)2=36+64=10
Half of 101010 is 555, so the radius is 555.
A circle has center (1,−3)(1, -3)(1,−3) and radius 13\sqrt{13}13. Which point lies on it?
Correct answer: B
A point is on the circle when (x−1)2+(y+3)2=13(x - 1)^2 + (y + 3)^2 = 13(x−1)2+(y+3)2=13. Test each candidate.
(4−1)2+(−1+3)2=32+22=9+4=13✓(4 - 1)^2 + (-1 + 3)^2 = 3^2 + 2^2 = 9 + 4 = 13 \checkmark(4−1)2+(−1+3)2=32+22=9+4=13✓
Only (4,−1)(4, -1)(4,−1) works; (0,0)(0, 0)(0,0) gives 101010, (5,0)(5, 0)(5,0) gives 252525, and (1,−3)(1, -3)(1,−3) is the center.
For what value of ccc does x2+y2−6x+4y+c=0x^2 + y^2 - 6x + 4y + c = 0x2+y2−6x+4y+c=0 describe a single point?
Complete the square: half of −6-6−6 is −3-3−3 (square 999), half of 444 is 222 (square 444).
(x−3)2+(y+2)2=13−c(x - 3)^2 + (y + 2)^2 = 13 - c(x−3)2+(y+2)2=13−c
A single point occurs when the right side is 000, so 13−c=013 - c = 013−c=0, giving c=13c = 13c=13.
A diameter of a circle has endpoints (−2,1)(-2, 1)(−2,1) and (4,9)(4, 9)(4,9). What is the center?
The center is the midpoint of the diameter, so average the endpoints' coordinates.
(−2+42,1+92)=(1,5)\left(\frac{-2 + 4}{2}, \frac{1 + 9}{2}\right) = (1, 5)(2−2+4,21+9)=(1,5)
The center is (1,5)(1, 5)(1,5).
Which point lies on the circle x2+y2=13x^2 + y^2 = 13x2+y2=13?
A point is on the circle when x2+y2=13x^2 + y^2 = 13x2+y2=13. Test each candidate.
22+32=4+9=13✓2^2 + 3^2 = 4 + 9 = 13 \checkmark22+32=4+9=13✓
Only (2,3)(2, 3)(2,3) works; (3,3)(3, 3)(3,3) gives 181818, and both (1,4)(1, 4)(1,4) and (4,1)(4, 1)(4,1) give 171717.
A diameter of a circle has endpoints (−2,1)(-2, 1)(−2,1) and (4,9)(4, 9)(4,9). What is the radius?
The radius is half the diameter's length. First find the distance between the endpoints.
d=(4−(−2))2+(9−1)2=36+64=10d = \sqrt{(4 - (-2))^2 + (9 - 1)^2} = \sqrt{36 + 64} = 10d=(4−(−2))2+(9−1)2=36+64=10
The circle (x−2)2+(y−2)2=r2(x - 2)^2 + (y - 2)^2 = r^2(x−2)2+(y−2)2=r2 passes through the origin. What is r2r^2r2?
Since the origin is on the circle, substitute (0,0)(0, 0)(0,0) into the equation.
(0−2)2+(0−2)2=4+4=8=r2(0 - 2)^2 + (0 - 2)^2 = 4 + 4 = 8 = r^2(0−2)2+(0−2)2=4+4=8=r2
So r2=8r^2 = 8r2=8 (and the radius itself is 222\sqrt{2}22).
Convert x2+y2+8x−6y=0x^2 + y^2 + 8x - 6y = 0x2+y2+8x−6y=0 to standard form. What are its center and radius?
The constant is already 000. Complete the square: half of 888 is 444 (square 161616), half of −6-6−6 is −3-3−3 (square 999).
(x+4)2+(y−3)2=0+16+9=25(x + 4)^2 + (y - 3)^2 = 0 + 16 + 9 = 25(x+4)2+(y−3)2=0+16+9=25
The center is (−4,3)(-4, 3)(−4,3) and the radius is 25=5\sqrt{25} = 525=5. (This circle passes through the origin.)
For x2+y2−10x+F=0x^2 + y^2 - 10x + F = 0x2+y2−10x+F=0 to be a circle of radius 444, what is FFF?
There is no yyy linear term. Complete the square on x2−10xx^2 - 10xx2−10x: half of −10-10−10 is −5-5−5 (square 252525).
(x−5)2+y2=25−F(x - 5)^2 + y^2 = 25 - F(x−5)2+y2=25−F
A radius of 444 needs 25−F=1625 - F = 1625−F=16, so F=9F = 9F=9.
A diameter of a circle has endpoints (2,−1)(2, -1)(2,−1) and (8,7)(8, 7)(8,7). Which is the circle's equation in standard form?
The center is the midpoint (2+82,−1+72)=(5,3)\left(\tfrac{2 + 8}{2}, \tfrac{-1 + 7}{2}\right) = (5, 3)(22+8,2−1+7)=(5,3). The diameter length is 62+82=10\sqrt{6^2 + 8^2} = 1062+82=10, so the radius is 555 and r2=25r^2 = 25r2=25.
(x−5)2+(y−3)2=25(x - 5)^2 + (y - 3)^2 = 25(x−5)2+(y−3)2=25
The center is (5,3)(5, 3)(5,3) with r2=25r^2 = 25r2=25.
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