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Equations of conic sections practice problems

Equations of parabolas, ellipses and hyperbolas from their foci, vertices and directrices. 51 problems with worked solutions; in 51 of them every equation is proved by a computer algebra system.

2 worked problems

Try each one before opening the solution. Every line is labeled with what checked it. How we verify

Problem 8.295

easy

Find the equation of the ellipse with vertices \( \displaystyle (\pm 6, 0) \) and foci \( \displaystyle (\pm 4, 0) \).

x²/a² + y²/b² = 1 with a the vertex distance and b² = a² − c².
Reviewed
\[ -16 + 6^{2} = 20 \]
b² = a² − c².✓ Proved
\[ \left. \frac{x^{2}}{36} + \frac{y^{2}}{20} \right|_{\substack{ x=6\\ y=0 }} = 1 \]
The vertex (6, 0) is on the curve.✓ Proved

Answer \( \frac{x^{2}}{36} + \frac{y^{2}}{20} = 1 \)

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

Problem 8.299

easy

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 4, 0) \) and foci \( \displaystyle (\pm 6, 0) \).

x²/a² − y²/b² = 1 with a the vertex distance and b² = c² − a².
Reviewed
\[ -16 + 6^{2} = 20 \]
b² = c² − a².✓ Proved
\[ \left. \frac{x^{2}}{16} - \frac{y^{2}}{20} \right|_{\substack{ x=4\\ y=0 }} = 1 \]
The vertex (4, 0) is on the curve.✓ Proved

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

✓ Nihil obstat Lines: 2 proved, 1 reviewed. The answer was also checked a second way, without looking at the solution. Reviewers found nothing wrong with the explanation. The full receipt · How we verify

More equations of conic sections practice

Find the equation of the ellipse with vertices \( \displaystyle (\pm 4, 0) \) and foci \( \displaystyle (\pm 1, 0) \).

Problem 8.296easy✓ Nihil obstat

Find the equation of the ellipse with vertices \( \displaystyle (\pm 6, 0) \) and foci \( \displaystyle (\pm 1, 0) \).

Problem 8.297easy✓ Every equation proved

Find the equation of the ellipse with vertices \( \displaystyle (\pm 5, 0) \) and foci \( \displaystyle (\pm 4, 0) \).

Problem 8.298easy✓ Nihil obstat

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

Problem 8.300easy✓ Every equation proved

Find the equation of the parabola with focus \( \displaystyle (-2, 0) \) and directrix \( \displaystyle x = 2 \).

Problem 8.301easy✓ Every equation proved

Find the equation of the parabola with focus \( \displaystyle (0, 1) \) and directrix \( \displaystyle y = -1 \).

Problem 8.302easy✓ Every equation proved

Find the equation of the ellipse with vertices \( \displaystyle (\pm 3, 0) \) and foci \( \displaystyle (\pm 1, 0) \).

Problem 8.303easy✓ Nihil obstat

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 4, 0) \) and foci \( \displaystyle (\pm 5, 0) \).

Problem 8.304easy✓ Nihil obstat

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 3, 0) \) and foci \( \displaystyle (\pm 7, 0) \).

Problem 8.305easy✓ Nihil obstat

Find the equation of the ellipse with vertices \( \displaystyle (\pm 3, 0) \) and foci \( \displaystyle (\pm 2, 0) \).

Problem 8.306easy✓ Nihil obstat

Find the equation of the parabola with focus \( \displaystyle (1, 0) \) and directrix \( \displaystyle x = -1 \).

Problem 8.307easy✓ Every equation proved

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 2, 0) \) and foci \( \displaystyle (\pm 3, 0) \).

Problem 8.308easy✓ Nihil obstat

Find the equation of the parabola with focus \( \displaystyle (0, 4) \) and directrix \( \displaystyle y = -4 \).

Problem 8.309easy✓ Every equation proved

Find the equation of the ellipse with vertices \( \displaystyle (\pm 5, 0) \) and foci \( \displaystyle (\pm 2, 0) \).

Problem 8.325easy✓ Every equation proved

Find the equation of the ellipse with vertices \( \displaystyle (\pm 4, 0) \) and foci \( \displaystyle (\pm 3, 0) \).

Problem 8.326easy✓ Nihil obstat

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 2, 0) \) and foci \( \displaystyle (\pm 5, 0) \).

Problem 8.327easy✓ Nihil obstat

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 4, 0) \) and foci \( \displaystyle (\pm 8, 0) \).

Problem 8.328easy✓ Nihil obstat

Find the equation of the ellipse with vertices \( \displaystyle (\pm 5, 0) \) and foci \( \displaystyle (\pm 1, 0) \).

Problem 8.329easy✓ Nihil obstat

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

Problem 8.330easy✓ Every equation proved

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 3, 0) \) and foci \( \displaystyle (\pm 5, 0) \).

Problem 8.331easy✓ Nihil obstat

Find the equation of the parabola with focus \( \displaystyle (-4, 0) \) and directrix \( \displaystyle x = 4 \).

Problem 8.332easy✓ Every equation proved

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 5, 0) \) and foci \( \displaystyle (\pm 6, 0) \).

Problem 8.333easy✓ Nihil obstat

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

Problem 8.334easy✓ Every equation proved

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 6, 0) \) and foci \( \displaystyle (\pm 7, 0) \).

Problem 8.417easy✓ Nihil obstat

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 6, 0) \) and foci \( \displaystyle (\pm 8, 0) \).

Problem 8.418easy✓ Nihil obstat

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 2, 0) \) and foci \( \displaystyle (\pm 6, 0) \).

Problem 8.419easy✓ Nihil obstat

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 5, 0) \) and foci \( \displaystyle (\pm 8, 0) \).

Problem 8.420easy✓ Nihil obstat

Find the equation of the ellipse with vertices \( \displaystyle (\pm 6, 0) \) and foci \( \displaystyle (\pm 3, 0) \).

Problem 8.421easy✓ Nihil obstat

Find the equation of the ellipse with vertices \( \displaystyle (\pm 2, 0) \) and foci \( \displaystyle (\pm 1, 0) \).

Problem 8.422easy✓ Every equation proved

Find the equation of the parabola with focus \( \displaystyle (-1, 0) \) and directrix \( \displaystyle x = 1 \).

Problem 8.423easy✓ Every equation proved

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 4, 0) \) and foci \( \displaystyle (\pm 7, 0) \).

Problem 8.424easy✓ Nihil obstat

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

Problem 8.425easy✓ Every equation proved

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 6, 0) \) and foci \( \displaystyle (\pm 10, 0) \).

Problem 8.426easy✓ Nihil obstat

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 3, 0) \) and foci \( \displaystyle (\pm 4, 0) \).

Problem 8.427easy✓ Nihil obstat

Find the equation of the ellipse with vertices \( \displaystyle (\pm 4, 0) \) and foci \( \displaystyle (\pm 2, 0) \).

Problem 8.428easy✓ Every equation proved

Find the equation of the ellipse with vertices \( \displaystyle (\pm 5, 0) \) and foci \( \displaystyle (\pm 3, 0) \).

Problem 8.429easy✓ Every equation proved

Find the equation of the parabola with focus \( \displaystyle (2, 0) \) and directrix \( \displaystyle x = -2 \).

Problem 8.430easy✓ Every equation proved

Find the equation of the parabola with focus \( \displaystyle (0, -1) \) and directrix \( \displaystyle y = 1 \).

Problem 8.431easy✓ Every equation proved

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 5, 0) \) and foci \( \displaystyle (\pm 7, 0) \).

Problem 8.477easy✓ Nihil obstat

Find the equation of the ellipse with vertices \( \displaystyle (\pm 6, 0) \) and foci \( \displaystyle (\pm 5, 0) \).

Problem 8.478easy✓ Nihil obstat

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 3, 0) \) and foci \( \displaystyle (\pm 6, 0) \).

Problem 8.479easy✓ Nihil obstat

Find the equation of the parabola with focus \( \displaystyle (0, -2) \) and directrix \( \displaystyle y = 2 \).

Problem 8.480easy✓ Every equation proved

Find the equation of the parabola with focus \( \displaystyle (0, -4) \) and directrix \( \displaystyle y = 4 \).

Problem 8.481easy✓ Every equation proved

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 2, 0) \) and foci \( \displaystyle (\pm 4, 0) \).

Problem 8.482easy✓ Nihil obstat

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 5, 0) \) and foci \( \displaystyle (\pm 9, 0) \).

Problem 8.483easy✓ Nihil obstat

Find the equation of the hyperbola with vertices \( \displaystyle (\pm 6, 0) \) and foci \( \displaystyle (\pm 9, 0) \).

Problem 8.484easy✓ Nihil obstat

Find the equation of the parabola with focus \( \displaystyle (4, 0) \) and directrix \( \displaystyle x = -4 \).

Problem 8.525easy✓ Every equation proved

Find the equation of the ellipse with vertices \( \displaystyle (\pm 6, 0) \) and foci \( \displaystyle (\pm 2, 0) \).

Problem 8.526easy✓ Every equation proved

Find the equation of the parabola with focus \( \displaystyle (0, 2) \) and directrix \( \displaystyle y = -2 \).

Problem 8.527easy✓ Every equation proved