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Conic Sections

18 practice questions with full worked solutions. Free, no account needed.

Problems & worked solutions

Problem #0182 International

Problem 1 Conic Sections

The center of the circle (x2)2+(y+3)2=16 is:

Show answer & worked solution
  1. A. (2,3)
  2. B. (2,3)
  3. C. (2,3)✓ correct
  4. D. (2,3)

Reading off: h=2, k=3. Center: (2,3).

Problem #0181 International

Problem 2 Conic Sections

The radius of the circle x2+y2=25 is:

Show answer & worked solution
  1. A. 5/2
  2. B. 5
  3. C. 5✓ correct
  4. D. 25

The standard form x2+y2=r2 gives r=25=5.

Problem #0183 International

Problem 3 Conic Sections

The vertex of the parabola y=(x1)2+3 is:

Show answer & worked solution
  1. A. (1,3)✓ correct
  2. B. (1,3)
  3. C. (1,3)
  4. D. (3,1)

y=(xh)2+k has vertex (h,k)=(1,3).

Problem #0933 US SAT

Problem 4 Circle

The equation of a circle with centre (2,3) and radius 5 is:

Show answer & worked solution
  1. A. (x2)2+(y+3)2=25✓ correct
  2. B. (x+2)2+(y3)2=25
  3. C. (x2)2+(y+3)2=5
  4. D. (x2)2(y+3)2=25

Centre (h,k)=(2,3) and r=5 (so r2=25):

(x2)2+(y(3))2=25    (x2)2+(y+3)2=25.

Problem #0185 International

Problem 5 Conic Sections

Find the radius of the circle x2+y24x+6y+4=0.

Show answer & worked solution
  1. A. 1
  2. B. 2
  3. C. 3✓ correct
  4. D. 4

(x24x)+(y2+6y)=4(x2)24+(y+3)29=4(x2)2+(y+3)2=9. Radius =3.

Problem #0187 International

Problem 6 Conic Sections

A circle centered at the origin passes through (3,4). Its equation is:

Show answer & worked solution
  1. A. x2+y2=5
  2. B. x2+y2=7
  3. C. x2+y2=25✓ correct
  4. D. (x3)2+(y4)2=25

r=9+16=5, so x2+y2=25.

Problem #0184 International

Problem 7 Conic Sections

For the ellipse x216+y29=1, the lengths of the semi-axes are:

Show answer & worked solution
  1. A. a=16b=9
  2. B. a=8b=6
  3. C. a=4b=3✓ correct
  4. D. a=2b=3

a2=16a=4; b2=9b=3.

Problem #0186 International

Problem 8 Conic Sections

For the hyperbola x29y216=1, the asymptotes have equations:

Show answer & worked solution
  1. A. y=±x
  2. B. y=±3x/4
  3. C. y=±4x/3✓ correct
  4. D. y=±5x/3

a=3, b=4, so asymptotes: y=±43x.

Problem #0188 International

Problem 9 Conic Sections

The parabola y2=8x has focus at:

Show answer & worked solution
  1. A. (0,2)
  2. B. (2,2)
  3. C. (2,0)✓ correct
  4. D. (8,0)

4p=8p=2. Focus: (2,0).

Problem #0935 US SAT

Problem 10 Ellipse

The ellipse x225+y29=1 has semi-major axis of length:

Show answer & worked solution
  1. A. 3
  2. B. 4
  3. C. 5✓ correct
  4. D. 9

Here a2=25 and b2=9, so a=5 and b=3.

The semi-major axis is max(a,b)=5.

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