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Chapter 6: Problem 55

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. $$25 x^{2}-10 x-200 y-119=0$$

Short Answer

Expert verified
The given equation describes a parabola.

Step by step solution

01

Rewrite in standard form

To express the given equation \(25 x^{2}-10 x-200 y-119=0\) in terms of y, treat x as a constant and rearrange the equation. This gives \[ y = \frac{25}{200}x^2-\frac{10}{200}x-\frac{119}{200} \] which simplifies to \[ y = \frac{1}{8}x^2-\frac{1}{20}x-\frac{119}{200} \]
02

Identify the conic section

From the equation, it can be seen that there is only one square term, meaning that there is only the \(x^{2}\) term and no \(y^{2}\) term. The coefficient of the \(x^{2}\) term is positive. These characteristics correspond to the standard form of a parabola.
03

Verify the result

To further confirm this conclusion, we can also see that the highest degree (power) of y is 1 (since there is no square term for y), which indicates a vertical axis of symmetry and further confirms that this is indeed a parabola.

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Most popular questions from this chapter

\(r=3 \cos 2 \theta\)

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