Understanding the Equation of an Ellipse

Oct 13, 2025 by windailys.com 63 views

Understanding the Equation of an Ellipse

The given equation represents an ellipse in the standard form, which is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ , where $(h,k)$ is the center of the ellipse, and $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively. In this case, the center of the ellipse is at $(5,4)$ , and the lengths of the semi-major and semi-minor axes are $25$ and $15$ , respectively.

Identifying the Major Axis

To determine the major axis of the ellipse, we need to compare the lengths of the semi-major and semi-minor axes. Since $a = 25$ is greater than $b = 15$ , the major axis is horizontal.

Determining the Directrix

The directrix of an ellipse is a line that is perpendicular to the major axis and is located at a distance of $c$ units from the center, where $c = \sqrt{a^2 - b^2}$ . In this case, $c = \sqrt{25^2 - 15^2} = \sqrt{625 - 225} = \sqrt{400} = 20$ .

Conclusion

Since the major axis is horizontal, the directrix of the ellipse is a vertical line that is 20 units from the center on the major axis.

Final Answer

The final answer is a vertical line that is 20 units.

Discussion

The equation of an ellipse is a fundamental concept in mathematics, and understanding the properties of an ellipse is crucial in various fields such as physics, engineering, and computer science. The directrix of an ellipse is an important concept that is used to determine the shape and size of the ellipse.

Key Takeaways

The equation of an ellipse is in the standard form $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ .
The center of the ellipse is at $(h,k)$ , and the lengths of the semi-major and semi-minor axes are $a$ and $b$ , respectively.
The major axis of the ellipse is horizontal if $a > b$ .
The directrix of the ellipse is a line that is perpendicular to the major axis and is located at a distance of $c$ units from the center, where $c = \sqrt{a^2 - b^2}$ .

Example Problems

Find the equation of an ellipse with a center at $(3,2)$ , a semi-major axis of length $10$ , and a semi-minor axis of length $5$ .
Determine the directrix of an ellipse with a center at $(0,0)$ , a semi-major axis of length $12$ , and a semi-minor axis of length $8$ .

Solutions

The equation of the ellipse is $\frac{(x-3)^2}{100} + \frac{(y-2)^2}{25} = 1$ .
The directrix of the ellipse is a vertical line that is $4$ units from the center on the major axis.

Conclusion

In conclusion, the equation of an ellipse is a fundamental concept in mathematics, and understanding the properties of an ellipse is crucial in various fields such as physics, engineering, and computer science. The directrix of an ellipse is an important concept that is used to determine the shape and size of the ellipse.

Frequently Asked Questions

Q: What is the standard form of the equation of an ellipse?

A: The standard form of the equation of an ellipse is $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ , where $(h,k)$ is the center of the ellipse, and $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively.

Q: How do I determine the major axis of an ellipse?

A: To determine the major axis of an ellipse, you need to compare the lengths of the semi-major and semi-minor axes. If $a > b$ , the major axis is horizontal. If $b > a$ , the major axis is vertical.

Q: What is the directrix of an ellipse?

A: The directrix of an ellipse is a line that is perpendicular to the major axis and is located at a distance of $c$ units from the center, where $c = \sqrt{a^2 - b^2}$ .

Q: How do I find the equation of an ellipse?

A: To find the equation of an ellipse, you need to know the center, semi-major axis, and semi-minor axis. You can use the standard form of the equation of an ellipse: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ .

Q: What is the difference between a semi-major axis and a semi-minor axis?

A: The semi-major axis is the length of the longest radius of the ellipse, while the semi-minor axis is the length of the shortest radius of the ellipse.

Q: How do I determine the distance between the center and the directrix?

A: To determine the distance between the center and the directrix, you need to calculate $c = \sqrt{a^2 - b^2}$ .

Q: What is the significance of the directrix in an ellipse?

A: The directrix of an ellipse is an important concept that is used to determine the shape and size of the ellipse. It is also used in various fields such as physics, engineering, and computer science.

Q: Can an ellipse have a directrix that is a horizontal line?

A: No, an ellipse cannot have a directrix that is a horizontal line. The directrix of an ellipse is always a line that is perpendicular to the major axis.

Q: Can an ellipse have a directrix that is a vertical line?

A: Yes, an ellipse can have a directrix that is a vertical line. This occurs when the major axis of the ellipse is vertical.

Q: How do I graph an ellipse?

A: To graph an ellipse, you need to know the center, semi-major axis, and semi-minor axis. You can use the standard form of the equation of an ellipse: $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$ . Plot the center and the points on the ellipse that are at a distance of $a$ and $b$ from the center.

Q: What is the equation of an ellipse with a center at $(3,2)$ , a semi-major axis of length $10$ , and a semi-minor axis of length $5$ ?

A: The equation of the ellipse is $\frac{(x-3)^2}{100} + \frac{(y-2)^2}{25} = 1$ .

Q: What is the directrix of an ellipse with a center at $(0,0)$ , a semi-major axis of length $12$ , and a semi-minor axis of length $8$ ?

A: The directrix of the ellipse is a vertical line that is $4$ units from the center on the major axis.