Find the standard form of the equation of the ellipse with the given characteristics and center at the origin.

Question

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. ​  ​

Quizplus · Accepted Answer

To find the standard form of the equation of an ellipse centered at the origin, we need to use the general form of the equation of an ellipse:

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

where $a$ is the length of the horizontal axis (major axis) and $b$ is the length of the vertical axis (minor axis).

From the given characteristics, we can see that:

- The length of the major axis is 18
- The length of the minor axis is 8
- The center is at the origin
- The ellipse is wider than it is tall (since $a > b$)
- We need to find the value of $e$, the eccentricity

To find $e$, we can use the formula:

$$e = \sqrt{1 - \frac{b^2}{a^2}}$$

Substituting in our values, we get:

$$e = \sqrt{1 - \frac{8^2}{18^2}} = \frac{2\sqrt{10}}{9}$$

Now we can write the standard form of the equation:

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

Substituting in $a = 9\sqrt{2}$, $b = 4\sqrt{2}$, and $e = \frac{2\sqrt{10}}{9}$, we get:

$$\frac{x^2}{(9\sqrt{2})^2} + \frac{y^2}{(4\sqrt{2})^2} = 1 - \left(\frac{2\sqrt{10}}{9}\right)^2$$

Simplifying, we get:

$$\frac{x^2}{162} + \frac{y^2}{32} = \frac{304}{243}$$

Multiplying both sides by 243, we get:

$$27x^2 + 81y^2 = 304$$

which is the standard form of the equation. Therefore, the answer is C.

Find the Standard Form of the Equation of the Ellipse