Logo The School of Mathematics
The School of Mathematics
ACT Math Circles, Ellipses, and Hyperbolas: 18 Practice Problems with Step-by-Step Explanations

Free 2,500+ ACT Math Practice Problems

Start Practicing →

ACT Math Circles, Ellipses, and Hyperbolas: 18 Practice Problems with Step-by-Step Explanations

Conic sections — circles, ellipses, and hyperbolas — all share a family resemblance on the ACT: each has a standard-form equation built around a center point, and once you can read that equation (or build one from a description), most problems become straightforward substitution. A circle's equation, (x−h)2+(y−k)2=r2, has center (h,k) and radius r. Note the equation uses r2, not r — a frequent source of careless errors when a problem gives the radius directly.

Beyond just reading off the center and radius, the ACT tests circles through diameter endpoints (the center is the midpoint of the diameter, and the radius is half the diameter's length, found with the distance formula), translations (shifting a circle's center without changing its radius), and equations that need to be rearranged first by completing the square. Ellipses extend the same idea with two different denominators, (x−h)2/a2 + (y−k)2/b2 = 1, where the larger denominator marks the direction of the major axis. Hyperbolas look similar but with a minus sign instead of a plus sign, and open along whichever variable is added (not subtracted).

Common traps include mixing up the sign in the center — (x−3)2 means the center's x-coordinate is +3, not −3 — confusing radius with diameter or with radius squared, and on ellipse and hyperbola problems, misreading which denominator corresponds to which axis. Work through the 18 problems below, including one that references a graph, then click to reveal each step-by-step explanation.

On this page:

  1. Problem 1 — Center of a circle
  2. Problem 2 — Diameter of a circle
  3. Problem 3 — Writing a circle's equation
  4. Problem 4 — Area of a circle
  5. Problem 5 — Radius from diameter endpoints
  6. Problem 6 — Writing a circle's equation
  7. Problem 7 — Modeling a circular path
  8. Problem 8 — Writing a circle's equation
  9. Problem 9 — Translating a circle
  10. Problem 10 — Describing points on a circle
  11. Problem 11 — Circle from a diameter length
  12. Problem 12 — Center by completing the square
  13. Problem 13 — Points of intersection
  14. Problem 14 — Circle from diameter endpoints
  15. Problem 15 — Center from a chord
  16. Problem 16 — Point on an ellipse
  17. Problem 17 — Endpoint of an ellipse's major axis
  18. Problem 18 — Identifying a hyperbola from a graph

Practice Problems

Problem 1

In the standard (x,y) plane, what is the center of the circle (x−3)2+(y+4)2=16?

  • A) (−3,4)
  • B) (3,4)
  • C) (3,−4)
  • D) (4,4)
Correct Answer: C

In the standard form (x−h)2+(y−k)2=r2, the center is (h,k). Here, x−3 means h=3, and y+4 = y−(−4) means k=−4.

The center is (3,−4).

Problem 2

In the standard (x,y) plane, what is the diameter of the circle (x+5)2+(y−3)2=36?

  • A) 6
  • B) 6π
  • C) 12
  • D) 12π
Correct Answer: C

Since r2=36, the radius is r=6.

The diameter is twice the radius: 2(6) = 12.

Problem 3

Which of the following equations is that of a circle that is in the standard (x,y) coordinate plane, has a center at (4,−7), and has a radius of 8?

  • A) (x+4)2+(y−7)2=8
  • B) (x−4)2+(y+7)2=8
  • C) (x+4)2+(y−7)2=64
  • D) (x−4)2+(y+7)2=64
Correct Answer: D

For center (4,−7), the equation needs (x−4)2+(y−(−7))2, which simplifies to (x−4)2+(y+7)2.

The right side must be radius squared: 82=64, not 8 itself.

Problem 4

In the standard (x,y) coordinate plane, what is the area of the following circle?

(x+2)2+(y−3)2=144

  • A) 12π
  • B) 64π
  • C) 144
  • D) 144π
Correct Answer: D

Since r2=144, the radius is r=12.

Area = πr2 = π(144) = 144π.

Problem 5

In the (x,y) coordinate plane, what is the radius of a circle having points (−2,1) and (4,9) as endpoints of a diameter?

  • A) 5
  • B) 6
  • C) 8
  • D) 10
Correct Answer: A

Find the diameter's length using the distance formula: √((4−(−2))2+(9−1)2) = √(36+64) = √100 = 10.

The radius is half the diameter: 10/2 = 5.

Problem 6

Which of the following equations is that of a circle that is in the standard (x,y) coordinate plane, has a center at (−11,5), and has a radius of 4?

  • A) (x+11)2+(y−5)2=4
  • B) (x−11)2+(y+5)2=4
  • C) (x−11)2+(y+5)2=2
  • D) (x+11)2+(y−5)2=16
Correct Answer: D

For center (−11,5), the equation needs (x−(−11))2+(y−5)2, which simplifies to (x+11)2+(y−5)2.

The right side must be radius squared: 42=16.

Problem 7

Michelle is standing at the center of a field swinging a ball attached to a 12-foot string. To graph the path of the ball on the (x,y) coordinate plane, Michelle decides the point where she is standing is the origin and each coordinate unit represents 1 foot. Which of the following equations best models the path of the ball during 1 full swing?

  • A) x2+y2=144
  • B) x2+y2=12
  • C) x2+y2=√12
  • D) x2/12 + y2/8 = 1
Correct Answer: A

The ball traces a circle centered at the origin (where Michelle stands) with radius equal to the string's length, 12 feet.

Using x2+y2=r2: x2+y2=122=144.

Problem 8

A circle in the standard (x,y) coordinate plane has a radius of 2√3 and a center at (0,5). Which of the following is the equation of the circle?

  • A) x2+(y+5)2=2√3
  • B) x2+(y−5)2=2√3
  • C) x2+(y+5)2=12
  • D) x2+(y−5)2=12
Correct Answer: D

For center (0,5): the equation needs (y−5)2, not (y+5)2.

The right side must be radius squared: (2√3)2 = 4(3) = 12, not 2√3 itself.

Problem 9

Circle A has a center at (6,−2) and a radius of 2. Circle B is formed by moving Circle A down 6 units and to the left by 3 units. Which of the following gives the correct equation for Circle B?

  • A) (x−3)2+(y+8)2=4
  • B) (x−3)2+(y−1)2=4
  • C) x2+(y+8)2=4
  • D) (x+3)2+(y−4)2=4
Correct Answer: A

Moving down 6 and left 3 shifts the center from (6,−2) to (6−3, −2−6) = (3,−8). The radius stays the same, 2.

New equation: (x−3)2+(y−(−8))2=22, which simplifies to (x−3)2+(y+8)2=4.

Problem 10

In the standard (x,y) coordinate plane, the circle centered at the origin that passes through (8,6) is the set of all points that are:

  • A) 10 coordinate units from the origin
  • B) 10 coordinate units from the origin and (8,6)
  • C) Equidistant from the origin and (8,6)
  • D) Equidistant from the line segment with endpoints at the origin and (8,6)
Correct Answer: A

The radius is the distance from the center (origin) to the point (8,6) on the circle: √(82+62) = √(64+36) = √100 = 10.

Every point on a circle is, by definition, exactly the radius distance from the center — so every point on this circle is 10 coordinate units from the origin.

Problem 11

A circle in the standard (x,y) coordinate plane contains two points that are 10 coordinate units apart and make a diameter. Which of the following could be the equation of the circle?

  • A) (x−10)2+(y−10)2=100
  • B) (x+100)2+(y−30)2=10
  • C) (x+√10)2+(y−√10)2=10
  • D) (x−95)2+(y−107)2=25
Correct Answer: D

A diameter of 10 means a radius of 5, so r2=25. Check each option's right-hand side: A gives r2=100 (radius 10, too large). B and C both give r2=10 (radius √10, too small).

Only D has r2=25, matching a radius of 5 and diameter of 10.

Problem 12

The equation of a circle in the xy-plane is shown below. What are the coordinates of the center of the circle?

x2+4x+y2−8y+5=0

  • A) (2,−4)
  • B) (−2,4)
  • C) (4,−8)
  • D) (−14,8)
Correct Answer: B

Complete the square for each variable. For x: x2+4x becomes (x+2)2−4. For y: y2−8y becomes (y−4)2−16.

Substituting back: (x+2)2−4+(y−4)2−16+5=0, so (x+2)2+(y−4)2=15. The center is (−2,4).

Problem 13

Suppose the equations (x−10)2+y2=16 and (x−5)2+y2=100 are graphed in the same standard (x,y) coordinate plane. How many points of intersection do these graphs share?

  • A) 0
  • B) 1
  • C) 2
  • D) 3
Correct Answer: A

The first circle has center (10,0) and radius 4. The second has center (5,0) and radius 10. The distance between the centers is |10−5|=5.

Since the distance between centers (5) plus the smaller radius (4) equals 9, which is still less than the larger radius (10), the smaller circle lies entirely inside the larger circle without touching its boundary. The circles don't intersect at all: 0 points.

Problem 14

In the (x,y) coordinate plane, the points (−4,3) and (4,−3) are the endpoints of the diameter of a circle. Which of the following is the equation of the circle?

  • A) x2+y2=25
  • B) x2+y2=100
  • C) (x−4)2+(y+3)2=25
  • D) (x−4)2+(y+3)2=100
Correct Answer: A

The center is the midpoint of the diameter: ((−4+4)/2, (3+(−3))/2) = (0,0), the origin.

The radius is half the diameter's length: √((4−(−4))2+(−3−3)2)/2 = √(64+36)/2 = 10/2 = 5, so r2=25. Equation: x2+y2=25.

Problem 15

A circle in the standard (x,y) coordinate plane intersects the y-axis at (0,5) and (0,11). The radius of the circle is 5 coordinate units. Which of the following could be the center of the circle?

I. (−4,8)    II. (0,8)    III. (4,8)

  • A) I only
  • B) II only
  • C) I and III only
  • D) I, II, and III
Correct Answer: C

The chord connecting (0,5) and (0,11) has length 6, and its midpoint is (0,8). The center must lie on the perpendicular bisector of this chord (the horizontal line y=8), at some horizontal distance d from the midpoint.

Using the right-triangle relationship between the radius, the half-chord, and d: d2+32=52, so d2=16 and d=4. The center is at (−4,8) or (4,8) — not (0,8), since that would require d=0 with a radius equal to only the half-chord length (3), not 5.

Problem 16

The point (3,b) lies on the ellipse with the given equation of (x+3)2/36 + (y+4)2/25 = 1. What is the value of b?

  • A) −4
  • B) −3
  • C) 1
  • D) 4
Correct Answer: A

Substitute x=3: (3+3)2/36 + (b+4)2/25 = 1, which simplifies to 36/36 + (b+4)2/25 = 1, or 1 + (b+4)2/25 = 1.

This means (b+4)2/25 = 0, so b+4=0, giving b=−4.

Problem 17

Which of the following points represents an endpoint of the major axis of the ellipse with the equation (x−3)2/9 + (y+4)2/25 = 1?

  • A) (3,−4)
  • B) (3,−29)
  • C) (3,1)
  • D) (0,4)
Correct Answer: C

Since the denominator under the y-term (25) is larger than the denominator under the x-term (9), the major axis is vertical. The center is (3,−4), and the semi-major axis length is √25=5.

The major axis endpoints are 5 units above and below the center: (3, −4+5)=(3,1) and (3, −4−5)=(3,−9). Of the choices given, (3,1) matches.

Problem 18

A hyperbola is shown below in the standard (x,y) coordinate plane. The hyperbola has which of the following equations?

  • A) (y−2)2/9 − x2/16 = 1
  • B) (y−2)2/36 − x2/16 = 1
  • C) y2/9 − (x−2)2/16 = 1
  • D) y2/36 − (x−2)2/16 = 1
Correct Answer: A

The graph shows a hyperbola opening vertically (up and down) with vertices at (0,5) and (0,−1). The center is the midpoint of the vertices: (0, (5+(−1))/2) = (0,2).

Since the vertices lie on the y-axis (x=0), only the y-term should be shifted, not the x-term — ruling out options C and D, which shift x instead. The distance from the center to each vertex is a=5−2=3, so a2=9, matching option A's denominator of 9 under the y-term.

Want more? Practice 2,500+ free ACT Math questions with instant explanations.

Try the Free ACT Math QBank →