Conic Sections: Parabolas, Ellipses and Hyperbolas: Precalculus Math Worksheet – PDF Download

Problem 1: Write the equation of a circle

Write the equation of the circle with the given center and radius.

1. center \((0,\ 0)\), radius \(7\)
2. center \((3,\ -2)\), radius \(5\)
3. center \((-4,\ 1)\), radius \(\sqrt{7}\)

Problem 2: Center and radius from an equation

State the center and radius of each circle.

1. \((x+4)^{2}+(y-1)^{2}=49\)
2. \(x^{2}+(y-5)^{2}=36\)
3. \((x-2)^{2}+(y+3)^{2}=20\)

Problem 3: Circle through a point

A circle has its center at \(C(1,\ 2)\) and passes through \(P(4,\ 6)\). Find its equation.

Problem 4: Circle from a diameter

The segment from \(A(-2,\ 3)\) to \(B(6,\ 7)\) is a diameter of a circle. Write the equation of the circle.

Problem 5: Complete the square: circles

Write each equation in standard form and name the center and radius.

1. \(x^{2}+y^{2}-8x+6y+9=0\)
2. \(x^{2}+y^{2}+10x-4y+13=0\)
3. \(x^{2}+y^{2}+2x-6y-15=0\)

Problem 6: Parabola from focus and directrix

Find the equation of the parabola with focus \((0,\ 3)\) and directrix \(y=-3\).

Problem 7: Focus and directrix from an equation

For each parabola, find \(p\), the focus and the directrix.

1. \(y^{2}=16x\)
2. \(x^{2}=-20y\)

Problem 8: Equal distances

Show that \(P(4,\ 2)\) lies on the parabola with focus \(F(0,\ 2)\) and directrix \(y=-2\) by comparing two distances.

Problem 9: Parabola with a shifted vertex

A parabola has vertex \((2,\ 1)\) and focus \((2,\ 4)\). Write its equation and its directrix.

Problem 10: Parabola opening left

Find the equation of the parabola with focus \((-3,\ 0)\) and directrix \(x=3\).

Problem 11: Ellipse features

For \(\frac{x^{2}}{25}+\frac{y^{2}}{9}=1\), find \(a\), \(b\), \(c\), the vertices, the co-vertices and the foci.

Problem 12: A vertical ellipse

For \(\frac{x^{2}}{16}+\frac{y^{2}}{25}=1\), find the vertices and the foci. Which axis is the major axis?

Problem 13: Ellipse from its foci

An ellipse has foci \((\pm 3,\ 0)\). The sum of the distances from any point on it to the foci is \(10\). Write its equation.

Problem 14: A shifted ellipse

For \(\frac{(x-2)^{2}}{36}+\frac{(y+1)^{2}}{20}=1\), find the center, vertices, foci, and the sum of the distances from a point on the ellipse to the foci.

Problem 15: Hyperbola features

For \(\frac{x^{2}}{16}-\frac{y^{2}}{9}=1\), find the vertices, foci and asymptotes.

Problem 16: A vertical hyperbola

For \(\frac{y^{2}}{9}-\frac{x^{2}}{16}=1\), find the vertices, foci and asymptotes.

Problem 17: Hyperbola from its foci

A hyperbola has foci \((\pm 5,\ 0)\). The absolute difference of the distances from any point on it to the foci is \(8\). Write its equation.

Problem 18: Identify an ellipse

Complete the square to write \(4x^{2}+9y^{2}-16x+18y-11=0\) in standard form. Name the conic, its center, and its foci.

Problem 19: Identify a hyperbola

Complete the square to write \(x^{2}-4y^{2}+6x+16y-23=0\) in standard form. Name the conic, its center, and its asymptotes.

Problem 20: Identify a parabola

Complete the square to write \(y^{2}-6y-8x+25=0\) in standard form. Find the vertex, focus and directrix.

Problem 21: Satellite dish

A satellite dish has a parabolic cross-section 8 feet wide and 2 feet deep. Place the vertex at the origin with the dish opening upward. Write an equation for the cross-section and find how far above the vertex the receiver should sit (at the focus).

Problem 22: Whispering gallery

A whispering gallery is an ellipse-shaped hall 100 feet long and 60 feet wide. Write its equation with the center at the origin and the long axis along the \(x\)-axis. Then find how far apart the two focus points are.

Problem 23: Sprinkler coverage

A sprinkler at \((20,\ 30)\) waters a circular region of radius \(15\) feet (coordinates in feet). Write the equation of the boundary. Is the flower at \((24,\ 38)\) watered? Is the bench at \((35,\ 45)\) watered?

Problem 24: Navigation stations

Two radio stations at \((-5,\ 0)\) and \((5,\ 0)\), measured in miles, send signals. A ship is exactly \(6\) miles closer to one station than to the other, so it lies on a hyperbola with these foci. Write the equation of the hyperbola, then find the \(y\)-coordinate of the ship when \(x=6\) and \(y>0\).

Problem 25: Elliptical garden

Priya outlines an elliptical garden 24 feet long and 18 feet wide by looping a string around two stakes and keeping it taut with a pencil. How far apart should the stakes be, and how long must the loop be around the stakes? Give the stake distance to the nearest hundredth.

Problem 26: Circular fountain

The edge of a circular fountain is modeled by \(x^{2}+y^{2}-12x-16y+75=0\), in feet, with the entrance at the origin. Find the center and radius, then find the shortest distance from the entrance to the edge of the fountain.

See every chapter on the Precalculus math page.