The Given Equation of the Ellipse

Oct 13, 2025 by talynt.org 53 views

Introduction to Ellipses

An ellipse is a fundamental concept in mathematics, particularly in geometry and algebra. It is a closed curve on a plane surrounding two focal points such that the sum of the distances to the two focal points is constant. The equation of an ellipse can be represented in various forms, but the standard form is given by:

$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

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.

The Given Equation of the Ellipse

The given equation of the ellipse is:

$\frac{(x-4)^2}{8} + \frac{(y-2)^2}{18} = 1$

From this equation, we can identify the center of the ellipse as $(4, 2)$ , the semi-major axis as $\sqrt{18} = 3\sqrt{2}$ , and the semi-minor axis as $\sqrt{8} = 2\sqrt{2}$ .

Properties of the Ellipse

The given ellipse has several properties that can be determined from its equation:

Center: The center of the ellipse is at the point $(4, 2)$ .
Major Axis: The major axis of the ellipse is along the line $y = 2$ , which means that the ellipse is taller in the vertical direction.
Minor Axis: The minor axis of the ellipse is along the line $x = 4$ , which means that the ellipse is wider in the horizontal direction.
Vertices: The vertices of the ellipse are the points where the ellipse intersects the major axis. In this case, the vertices are at $(4, 2 + 3\sqrt{2})$ and $(4, 2 - 3\sqrt{2})$ .

Vertices of the Ellipse

The vertices of the ellipse are the points where the ellipse intersects the major axis. In this case, the vertices are at $(4, 2 + 3\sqrt{2})$ and $(4, 2 - 3\sqrt{2})$ . These points can be obtained by substituting the value of $y$ in the equation of the ellipse.

Discussion on the Vertices

Conclusion

In conclusion, the given equation of the ellipse is $\frac{(x-4)^2}{8} + \frac{(y-2)^2}{18} = 1$ . The center of the ellipse is at the point $(4, 2)$ , the semi-major axis is $3\sqrt{2}$ , and the semi-minor axis is $2\sqrt{2}$ . The vertices of the ellipse are at $(4, 2 + 3\sqrt{2})$ and $(4, 2 - 3\sqrt{2})$ .

Introduction to Ellipses

$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

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&A on the Equation of an Ellipse

Q: What is the center of the ellipse?

A: The center of the ellipse is the point $(h, k)$ in the standard form of the equation of the ellipse. In the given equation, the center of the ellipse is at the point $(4, 2)$ .

Q: What is the semi-major axis of the ellipse?

A: The semi-major axis of the ellipse is the length of the semi-major axis, which is denoted by $a$ in the standard form of the equation of the ellipse. In the given equation, the semi-major axis is $\sqrt{18} = 3\sqrt{2}$ .

Q: What is the semi-minor axis of the ellipse?

A: The semi-minor axis of the ellipse is the length of the semi-minor axis, which is denoted by $b$ in the standard form of the equation of the ellipse. In the given equation, the semi-minor axis is $\sqrt{8} = 2\sqrt{2}$ .

Q: What are the vertices of the ellipse?

A: The vertices of the ellipse are the points where the ellipse intersects the major axis. In the given equation, the vertices are at $(4, 2 + 3\sqrt{2})$ and $(4, 2 - 3\sqrt{2})$ .

Q: How do I find the vertices of the ellipse?

A: To find the vertices of the ellipse, you need to substitute the value of $y$ in the equation of the ellipse. In this case, the vertices are at $(4, 2 + 3\sqrt{2})$ and $(4, 2 - 3\sqrt{2})$ .

Q: What is the major axis of the ellipse?

A: The major axis of the ellipse is the line that passes through the center of the ellipse and the vertices of the ellipse. In the given equation, the major axis is along the line $y = 2$ .

Q: What is the minor axis of the ellipse?

A: The minor axis of the ellipse is the line that passes through the center of the ellipse and the endpoints of the minor axis. In the given equation, the minor axis is along the line $x = 4$ .

Conclusion

In conclusion, the equation of an ellipse is a fundamental concept in mathematics, particularly in geometry and algebra. The standard form of the equation of an ellipse is given by:

$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

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. The given equation of the ellipse is $\frac{(x-4)^2}{8} + \frac{(y-2)^2}{18} = 1$ . The center of the ellipse is at the point $(4, 2)$ , the semi-major axis is $3\sqrt{2}$ , and the semi-minor axis is $2\sqrt{2}$ . The vertices of the ellipse are at $(4, 2 + 3\sqrt{2})$ and $(4, 2 - 3\sqrt{2})$ .