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Exercises Conic sections

Example 1 from Algebra and Trigonometry, 12.4 Rotation of Axes

Identifying a Conic from Its General Form

Identify the graph of each of the following nondegenerate conic sections.

ⓐ $4 x^{2} - 9 y^{2} + 36 x + 36 y - 125 = 0$
ⓑ $9 y^{2} + 16 x + 36 y - 10 = 0$
ⓒ $3 x^{2} + 3 y^{2} - 2 x - 6 y - 4 = 0$
ⓓ $- 25 x^{2} - 4 y^{2} + 100 x + 16 y + 20 = 0$

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Solution
- ⓐ Rewriting the general form, we have
  $A = 4$ and $C = −9,$ so we observe that $A$ and $C$ have opposite signs. The graph of this equation is a hyperbola.
- ⓑ Rewriting the general form, we have
  $A = 0$ and $C = 9.$ We can determine that the equation is a parabola, since $A$ is zero.
- ⓒ Rewriting the general form, we have
  $A = 3$ and $C = 3.$ Because $A = C,$ the graph of this equation is a circle.
- ⓓ Rewriting the general form, we have
  $A = −25$ and $C = −4.$ Because $A C > 0$ and $A \neq C,$ the graph of this equation is an ellipse.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Identify the graph of each of the following nondegenerate conic sections.

ⓐ $16 y^{2} - x^{2} + x - 4 y - 9 = 0$
ⓑ $16 x^{2} + 4 y^{2} + 16 x + 49 y - 81 = 0$

How did it go?

PreviousSolving Applied Problems Involving Parabolas Next Finding a New Representation of an Equation after Rotating through a Given Angle

From Algebra and Trigonometry (1st edition) by OpenStax (Rice University), section 12.4, licensed under CC BY 4.0. Original at openstax.org. We split the solution into steps; the words, numbers and figures are the book's. OpenStax doesn't endorse this site.