Determine whether the transverse axis and foci of the hyperbola are on the -axis or the -axis.

The transverse axis and foci of the hyperbola are on the -axis.
The transverse axis and foci of the hyperbola are on the -axis.

Ask by Schultz Park. in the United States

Mar 25,2025

Question

Determine whether the transverse axis and foci of the hyperbola are on the -axis or the -axis.

The transverse axis and foci of the hyperbola are on the -axis.
The transverse axis and foci of the hyperbola are on the -axis.

Ask by Schultz Park. in the United States

Mar 25,2025

UpStudy AI · Accepted Answer

To determine whether the transverse axis and foci of the hyperbola given by the equation

$$
-x^{2} + \frac{y^{2}}{20} = 1
$$

are on the $$ x $$-axis or the $$ y $$-axis, we first need to rewrite the equation in standard form.

1. **Rearranging the equation**: 
   We can multiply the entire equation by -1 to make the $$ x^2 $$ term positive:

$$
{x^{2}} - \frac{y^{2}}{20} = -1
   $$

This is not the standard form yet. We need to isolate the terms:

$$
\frac{y^{2}}{20} - x^{2} = 1
   $$

2. **Identifying the standard form**:
   The standard form of a hyperbola is given by:

$$
\frac{y^{2}}{a^{2}} - \frac{x^{2}}{b^{2}} = 1
   $$

In our case, we can see that:

- $$ a^{2} = 20 $$ (so $$ a = \sqrt{20} $$)
   - $$ b^{2} = 1 $$ (so $$ b = 1 $$)

3. **Determining the orientation**:
   Since the $$ y^{2} $$ term is positive and the $$ x^{2} $$ term is negative, this indicates that the hyperbola opens upwards and downwards. Therefore, the transverse axis is vertical.

4. **Conclusion**:
   The transverse axis and foci of the hyperbola are on the $$ y $$-axis.

Thus, the answer is: **The transverse axis and foci of the hyperbola are on the $$ y $$-axis.**
The transverse axis and foci of the hyperbola are on the $$ y $$-axis.

Determine whether the transverse axis and foci of the hyperbola are on the -axis or the -axis.

The transverse axis and foci of the hyperbola are on the -axis.
The transverse axis and foci of the hyperbola are on the -axis.

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Determine whether the transverse axis and foci of the hyperbola are on the -axis or the -axis. The transverse axis and foci of the hyperbola are on the -axis. The transverse axis and foci of the hyperbola are on the -axis.

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Determine whether the transverse axis and foci of the hyperbola are on the -axis or the -axis.

The transverse axis and foci of the hyperbola are on the -axis.
The transverse axis and foci of the hyperbola are on the -axis.