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

Question 1: Equations of circles (2 points)

1. Write the equation of the circle with center \((-3,\ 4)\) and radius \(6\). (1 point)
2. State the center and radius of \((x-1)^{2}+(y+5)^{2}=18\). (1 point)

Question 2: Completing the square for a circle (3 points)

A circle has equation \(x^{2}+y^{2}+6x-8y-11=0\).

1. Write the equation in standard form. (1 point)
2. Find the center and radius. (1 point)
3. Determine whether the point \((3,\ 4)\) lies on the circle. (1 point)

Question 3: Parabola from focus and directrix (3 points)

A parabola has focus \((0,\ -2)\) and directrix \(y=2\), as shown.

1. Write its equation. (2 points)
2. Find the \(y\)-coordinate of the point on the parabola with \(x=4\). (1 point)

Question 4: Parabola features (2 points)

For the parabola \(y^{2}=12x\), find the focus and the directrix. (1 point each)

Question 5: Ellipse features (3 points)

The graph of \(\frac{x^{2}}{49}+\frac{y^{2}}{24}=1\) is shown.

1. Find the vertices on the major axis. (1 point)
2. Find the foci. (1 point)
3. For any point on the ellipse, what is the sum of the distances to the foci? (1 point)

Question 6: Ellipse from its foci (2 points)

An ellipse has foci \((0,\ \pm 4)\). The sum of the distances from any point on it to the two foci is \(10\). Write its equation.

Question 7: Hyperbola features (3 points)

The graph of \(\frac{x^{2}}{9}-\frac{y^{2}}{16}=1\) is shown.

1. Find the vertices. (1 point)
2. Find the foci. (1 point)
3. Write the equations of the asymptotes. (1 point)

Question 8: Identify a conic (2 points)

Consider \(9x^{2}-16y^{2}-36x-32y-124=0\).

1. Write the equation in standard form. (1 point)
2. Name the conic and state its center. (1 point)

⬇ Download the Precalculus Conic Sections: Parabolas, Ellipses and Hyperbolas test (PDF, 89 KB)

See every chapter on the Precalculus math page.