Unit 8 test: Conic Sections

Find the focus of the parabola $y^2 = 16x$.

The directrix of the parabola $(x - 2)^2 = -12(y + 1)$ is the line $y = c$. Find $c$.

Which equation describes the parabola with focus $(0, 2)$ and directrix $y = -2$?

Find the vertex of the parabola $y^2 + 8y - 4x + 4 = 0$.

The ellipse $\dfrac{x^2}{49} + \dfrac{y^2}{24} = 1$ has its foci on the $x$-axis. Enter the $x$-coordinates of both foci.

Find the length of the major axis of the ellipse $16x^2 + 25y^2 - 32x + 100y - 284 = 0$.

An ellipse has center $(3, -1)$, a vertical major axis of length $8$, and a minor axis of length $6$. Which is its equation?

A bridge has an arch shaped like the top half of an ellipse. The arch is $40$ meters wide at its base and $12$ meters tall at its center. How tall is the arch $5$ meters from the center? Round to the nearest tenth of a meter.

Find the slopes of the asymptotes of $\dfrac{x^2}{4} - \dfrac{y^2}{49} = 1$.

Find the focus with the positive $y$-coordinate of the hyperbola $\dfrac{y^2}{9} - \dfrac{x^2}{40} = 1$.

A hyperbola centered at the origin has vertices $(\pm 2, 0)$ and foci $\left(\pm\sqrt{13}, 0\right)$. Its equation is $\dfrac{x^2}{4} - \dfrac{y^2}{b^2} = 1$. Find $b^2$.

Identify the conic: $3x^2 - 5y^2 + 6x + 20y - 32 = 0$. (Type circle, ellipse, parabola or hyperbola.)

Find the radius of the circle $x^2 + y^2 - 8x + 6y = 0$.

Use the discriminant to identify $x^2 - 6xy + 9y^2 + 2x - 3 = 0$.