Conic Sections practice games — free for Pre-Calculus.
This unit covers parabolas, ellipses, hyperbolas and identifying conics — essential concepts for Pre-Calculus. Use our interactive study games to test your understanding, or review questions in traditional format below.
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This unit covers parabolas, ellipses, hyperbolas and identifying conics — essential concepts for Pre-Calculus. Use our interactive study games to test your understanding, or review questions in traditional format below.
Key Concepts Breakdown
1 Parabolas
A parabola is the set of all points equidistant from a fixed point (focus) and a fixed line (directrix). Students must know both standard forms and be able to identify the vertex, axis of symmetry, focus, and directrix. Vertical parabolas open up or down; horizontal parabolas open left or right.
Key Points
- Vertical form: (x - h)² = 4p(y - k); opens up if p > 0, down if p < 0
- Horizontal form: (y - k)² = 4p(x - h); opens right if p > 0, left if p < 0
- Focus is |p| units from vertex along the axis; directrix is |p| units on the opposite side
- Vertex is always the midpoint between focus and directrix
Write the equation of a parabola with vertex (2, -3) and focus (2, 1).
The focus is directly above the vertex, so this is a vertical parabola with h = 2, k = -3. The value of p is the distance from vertex to focus: p = 1 - (-3) = 4. Substituting into (x - h)² = 4p(y - k) gives (x - 2)² = 16(y + 3).
2 Ellipses
An ellipse is the set of all points where the sum of distances to two fixed points (foci) is constant. Students must know the standard form, distinguish the major and minor axes, and locate the foci using the relationship c² = a² - b². The larger denominator always indicates the major axis direction.
Key Points
- Horizontal major axis: (x - h)²/a² + (y - k)²/b² = 1, where a > b > 0
- Vertical major axis: (x - h)²/b² + (y - k)²/a² = 1, where a > b > 0
- Foci lie on the major axis; c² = a² - b², so c < a always
- Vertices are at distance a from center; co-vertices are at distance b from center
Find the foci of the ellipse (x + 1)²/25 + (y - 2)²/9 = 1.
Since 25 > 9, the major axis is horizontal with a² = 25 and b² = 9. Using c² = a² - b² gives c² = 25 - 9 = 16, so c = 4. The foci are 4 units left and right of the center (-1, 2), giving foci at (-5, 2) and (3, 2).
3 Hyperbolas
A hyperbola is the set of all points where the absolute difference of distances to two fixed points (foci) is constant. Students must know both orientations, find the foci, and write the equations of the asymptotes. Unlike ellipses, c² = a² + b² for hyperbolas.
Key Points
- Horizontal transverse axis: (x - h)²/a² - (y - k)²/b² = 1; opens left and right
- Vertical transverse axis: (y - k)²/a² - (x - h)²/b² = 1; opens up and down
- Foci: c² = a² + b² (note the plus sign, different from ellipses)
- Asymptotes pass through center with slopes ±b/a (horizontal) or ±a/b (vertical)
Find the asymptotes of the hyperbola (y - 1)²/16 - (x + 3)²/9 = 1.
The positive term is under y², so this is a vertical hyperbola with center (-3, 1), a² = 16 (a = 4), and b² = 9 (b = 3). Asymptotes for a vertical hyperbola have slopes ±a/b = ±4/3. The asymptote equations are y - 1 = ±(4/3)(x + 3).
4 Identifying Conics
Students must be able to identify the type of conic from a general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 by examining the coefficients. The key test is comparing the coefficients of x² and y² after eliminating the xy term (B = 0 on most exams). Completing the square is required to convert general form to standard form.
Key Points
- Parabola: exactly one squared term (A = 0 or C = 0, but not both)
- Circle: both squared terms with equal coefficients (A = C, same sign)
- Ellipse: both squared terms with different positive coefficients (A ≠ C, same sign)
- Hyperbola: both squared terms with opposite signs (A and C have opposite signs)
Identify the conic: 4x² - 9y² + 16x + 18y - 29 = 0.
The coefficients of x² and y² are +4 and -9, which have opposite signs, so this is a hyperbola. To confirm the center, complete the square: 4(x² + 4x + 4) - 9(y² - 2y + 1) = 29 + 16 - 9, giving (x + 2)²/9 - (y - 1)²/4 = 1 with center (-2, 1).
Questions, answered.
What is Conic Sections?
Conic Sections is Unit 8 of Pre-Calculus, covering parabolas, ellipses, hyperbolas and identifying conics.
How to study for Pre-Calculus Unit 8?
Start with the Quick Summary above, review the Key Concepts, then test yourself with our interactive study games. Aim for 80%+ accuracy before moving on.
How many questions are in this unit?
This unit has 27+ review questions across 5 different game modes.