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Conic sections — ACT practice questions

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Practice 36 Conic sections questions in the app

What this topic is

Parabolas, circles, ellipses and hyperbolas are the curves in Conic sections, and each is recognized from its equation.

A hyperbola in standard form has a minus sign between its squared terms, while an ellipse has a plus sign and two different denominators. The ACT asks for features: the center, where (x + h) means a center at −h; the vertices; the slopes of a hyperbola's asymptotes; the foci; or the directrix of a parabola.

For a hyperbola, the focal distance c satisfies c squared equals a squared plus b squared, whereas for an ellipse it is a squared minus b squared. Choices often differ only in sign.

Sample questions

Pick an answer to see whether it is right — nothing is saved, and nothing needs an account.

Question 1Mid

Which of the following gives the center, vertices, and asymptote slopes of $\frac{(x + 1)^{2}}{16} - \frac{(y - 3)^{2}}{9} = 1$ ?

Center

(- 1, 3)

; vertices

(- 5, 3)

and

(3, 3)

; slopes

\pm \frac{4}{3}

✗ Not this oneCenter

(1, - 3)

; vertices

(- 3, - 3)

and

(5, - 3)

; slopes

\pm \frac{3}{4}

✗ Not this oneCenter

(- 1, 3)

; vertices

(- 4, 3)

and

(2, 3)

; slopes

\pm \frac{4}{3}

✗ Not this oneCenter

(- 1, 3)

; vertices

(- 5, 3)

and

(3, 3)

; slopes

\pm \frac{3}{4}

✓ Correct

Show answer

Answer: D — Center $(- 1, 3)$ ; vertices $(- 5, 3)$ and $(3, 3)$ ; slopes $\pm \frac{3}{4}$

The center is $(- 1, 3)$ , and the positive $x$ term gives a horizontal transverse axis with $a = 4$ . Thus, the vertices are $(- 5, 3)$ and $(3, 3)$ and the asymptote slopes are $\pm b / a = \pm \frac{3}{4}$ .

Question 2Mid

The parabola $y^{2} = 12 x$ is graphed in the standard $(x, y)$ coordinate plane. Which of the following is an equation of its directrix?

x = - 3

✓ Correct

x = 3

✗ Not this one

y = - 3

✗ Not this one

y = 3

✗ Not this one

Show answer

Answer: A — $x = - 3$

For $y^{2} = 4 p x$ with $4 p = 12$ , $p = 3$ . The parabola has vertex $(0, 0)$ and opens right, so its directrix is the vertical line $x = - 3$ .

Question 3Mid

In the standard $(x, y)$ coordinate plane, what are the coordinates of the foci of the hyperbola $\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1$ ?

(\pm 5, 0)

✓ Correct

(\pm 4, 0)

✗ Not this one

(\pm 3, 0)

✗ Not this one

(0, \pm 5)

✗ Not this one

Show answer

Answer: A — $(\pm 5, 0)$

For $\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1$ , $c^{2} = a^{2} + b^{2} = 16 + 9 = 25$ , so $c = 5$ . The transverse axis is horizontal, so the foci are $(\pm 5, 0)$ .

Every question is picked for where you are right now, wrong answers come back until they stick, and it all works offline.

Practice 36 Conic sections questions in the app

Conic sections — ACT practice questions

What this topic is

Sample questions

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Mathematics