Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
The last problems model dishes, halls, bridges, navigation and orbits. Give exact values and decimals where asked.
Standards: G-GPE.A.1, G-GPE.A.2, G-GPE.A.3 · Level: Grade 12 Precalculus
Problem 1 Circle from Its Equation ★★★
Find the center and radius of (x − 6)² + (y + 1)² = 20.
Problem 2 Circle from a Point ★★★
A circle is centered at (−7, 4) and its edge goes through (−1, −1). Give its equation in standard form.
Problem 3 Focus and Directrix ★★★
For the parabola x² = 12y, find the vertex, focus and directrix.
Problem 4 A Sideways Parabola ★★★
For y² = −20x, find the focus and directrix and state which way the parabola opens.
Problem 5 Equation from the Figure ★★★
The parabola in the figure has its vertex at the origin and its focus at (0, −4). Write its equation.
Problem 6 Parts of an Ellipse ★★★
For x²49 + y²25 = 1, find the vertices, the endpoints of the minor axis and the foci.
Problem 7 Ellipse from Vertices and Foci ★★★
An ellipse centered at the origin has vertices (0, ±9) and foci (0, ±5), as shown. Write its equation.
Problem 8 Parts of a Hyperbola ★★★
For x²25 − y²11 = 1, find the vertices, the foci and the equations of the asymptotes.
Problem 9 A Vertical Hyperbola ★★★
For y²9 − x²27 = 1, find the vertices, the foci and the asymptotes.
Problem 10 Shifted Parabola ★★★
A parabola has vertex (5, −2) and focus (5, 3). Write its equation and find its directrix.
Problem 11 Completing the Square: Circle ★★★
Rewrite x² + y² − 8x + 10y + 5 = 0 in standard form and give the center and radius.
Problem 12 Completing the Square: Parabola ★★★
Rewrite y² − 4y − 12x + 40 = 0 in standard form. Find the vertex, focus and directrix.
Problem 13 Completing the Square: Ellipse ★★★
Identify 9x² + 4y² − 18x + 16y − 11 = 0 and find its center, vertices and foci.
Problem 14 Completing the Square: Hyperbola ★★★
Rewrite 4x² − 5y² + 24x + 10y + 11 = 0 in standard form. Find the center, foci and asymptotes.
Problem 15 Satellite Dish ★★★
A satellite dish is 10 feet wide and 1.25 feet deep. Model its cross-section as a parabola with its vertex at the origin opening upward. (a) Write the equation. (b) How far above the vertex should the receiver be placed, at the focus?
Problem 16 Whispering Hall ★★★
A hall is shaped like an ellipse 80 feet long and 30 feet wide. Two people stand at the foci and whisper to each other. (a) Write the equation of the ellipse with its center at the origin. (b) How far from the center is each focus, to the nearest tenth of a foot? (c) How far apart are the two people?
Problem 17 Elliptical Bridge Arch ★★★
A bridge arch is the upper half of an ellipse 60 feet wide at the base and 18 feet high at the center. (a) Write an equation with the center of the base at the origin. (b) Find the height of the arch 20 feet from the center. (c) A truck 14 feet tall drives under the arch. How wide can the truck be if it is centered? Round to the nearest tenth.
Problem 18 Radio Station Hyperbola ★★★
Two radio stations are 180 miles apart. A ship measures that it is 100 miles closer to one station than the other. Place the stations at (±90, 0). (a) Write the equation of the hyperbola of possible ship positions. (b) Where does it cross the x-axis? (c) Find its asymptotes.
Problem 19 A Comet’s Orbit ★★★
A comet travels on an ellipse with the sun at one focus. The major axis is 52 million miles long, and at its closest the comet is 10 million miles from the sun. (a) Find a and c. (b) Find b² and write an equation with the center at the origin. (c) Find the greatest distance from the sun.
Problem 20 Identify and Describe ★★★
Show that 4x² − 9y² − 24x − 36y − 36 = 0 is a hyperbola. Give its standard form, center, vertices, foci and asymptotes.
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