Back to Exercise: Derive conic equations

Exercises: Derive Equations of Ellipses and Hyperbolas

Show your work for each problem. Assume every conic is centered at the origin unless the problem says otherwise.

Grade 10·21 problems·~35 min·Common Core Math - HS Geometry·standard·hsg-gpe-a-3

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Warm-Up: Review What You Know

These problems review skills you already have from your work on circles.

What is the distance between the points $(-3, 1)$ and $(5, 7)$ ?

$\sqrt{14}$

100

A circle is the set of all points in a plane at a fixed distance from one fixed point. Which description follows the same pattern but uses two fixed points and a constant sum?

The set of all points whose distances to the two fixed points differ by a constant

The set of all points whose distances to the two fixed points add up to a constant

The set of all points that are the same distance from both fixed points

The set of all points whose distance to one fixed point equals the distance to a fixed line

A right triangle has legs of length $b$ and $c$ and hypotenuse of length $a$ . If $a = 13$ and $c = 5$ , what is $b$ ?

Fluency Practice

For the ellipse $\frac{x^2}{169} + \frac{y^2}{144} = 1$ , find the coordinates of the two vertices (the endpoints of the major axis).

Where are the foci of the ellipse $\frac{x^2}{100} + \frac{y^2}{64} = 1$ ?

$(0, \pm 6)$

$(\pm \sqrt{164}, 0)$

$(\pm 10, 0)$

$(\pm 6, 0)$

For the hyperbola $\frac{x^2}{144} - \frac{y^2}{25} = 1$ , find the coordinates of the two foci.

$Coordinate plane showing a hyperbola with two branches opening left and right from (8, 0) and (-8, 0), with a dashed central rectangle and two dashed diagonal lines through the origin$

What are the equations of the asymptotes of the hyperbola $\frac{x^2}{64} - \frac{y^2}{36} = 1$ ?

$x = 0$ and $y = 0$

$y = \frac{3}{4}x$ and $y = -\frac{3}{4}x$

$y = \frac{4}{3}x$ and $y = -\frac{4}{3}x$

$x = 8$ and $x = -8$

Compute the eccentricity of the ellipse $\frac{x^2}{25} + \frac{y^2}{16} = 1$ . Express your answer as a fraction in simplest form.

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