Conic Sections Test – Precalculus Math (Printable PDF)

Question 1 Circle in General Form (2 points)

Find the center and radius of the circle x² + y² + 10x − 6y + 18 = 0.

Question 2 A Downward Parabola (2 points)

A parabola has vertex (−1, 4) and directrix y = 7. Write its equation and find its focus.

Question 3 Reading an Ellipse (3 points)

The figure shows x²20 + y²36 = 1.

1. Find the vertices. (1 point)
2. Find the endpoints of the minor axis. (1 point)
3. Find the foci. (1 point)

Question 4 A Hyperbola from Foci (3 points)

A hyperbola is centered at the origin with foci (±7, 0) and vertices (±4, 0).

1. Write its equation. (2 points)
2. Find the equations of its asymptotes. (1 point)

Question 5 Identify the Conic (2 points)

Complete the square to identify 4x² + y² − 24x + 4y + 24 = 0. State the center and the foci.

Question 6 Parabolic Mirror (2 points)

A parabolic mirror is 24 centimeters wide and 4 centimeters deep, with its vertex at the origin opening upward. Write its equation and find how far the light source belongs from the vertex.

Question 7 Running Track (3 points)

An elliptical running track is 120 meters long and 80 meters wide.

1. Write an equation with the center at the origin and the long side horizontal. (1 point)
2. Find the distance from the center to each focus, to the nearest tenth. (1 point)
3. Find the distance between the two foci. (1 point)

Question 8 Checking the Definition (3 points)

An ellipse is given by x²49 + y²40 = 1.

1. Find its foci. (1 point)
2. Show that the point (0, 2√10) is on the ellipse. (1 point)
3. Find the distances from that point to each focus and show that their sum is 2a. (1 point)