What Are The Foci Of The Ellipse $9x^2 + 5y^2 - 45 = 0$?Write Your Answer In Simplified, Rationalized Form.

Introduction

In mathematics, an ellipse is a fundamental concept 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 in standard form is given by $\frac{(x-h)^2}{a2} + \frac{(y-k)^2}{b2} = 1$, where $(h,k)$ is the center of the ellipse, and $a$ and $b$ are the semi-major and semi-minor axes, respectively. However, not all ellipse equations are in standard form. In this article, we will explore how to find the foci of an ellipse given by the equation $9x^2 + 5y^2 - 45 = 0$.

Understanding the Equation

To find the foci of the ellipse, we first need to rewrite the given equation in standard form. The equation $9x^2 + 5y^2 - 45 = 0$ can be rewritten as $\frac{9x^2}{45} + \frac{5y^2}{45} = 1$, which simplifies to $\frac{x^2}{5} + \frac{y^2}{9} = 1$. Comparing this equation with the standard form of an ellipse, we can see that the center of the ellipse is at the origin $(0,0)$, and the semi-major and semi-minor axes are $a = 3$ and $b = \sqrt{5}$, respectively.

Finding the Foci

The foci of an ellipse are located on the major axis, which is the x-axis in this case. The distance between the center of the ellipse and each focus is given by the formula $c = \sqrt{a^2 - b^2}$, where $c$ is the distance from the center to each focus. Plugging in the values of $a$ and $b$, we get $c = \sqrt{3^2 - (\sqrt{5})^2} = \sqrt{9 - 5} = \sqrt{4} = 2$.

Simplified and Rationalized Form

Since the distance from the center to each focus is $c = 2$, the coordinates of the foci are $(\pm 2, 0)$. Therefore, the foci of the ellipse are located at $(2, 0)$ and $(-2, 0)$.

Conclusion

In conclusion, we have found the foci of the ellipse given by the equation $9x^2 + 5y^2 - 45 = 0$. The foci are located at $(2, 0)$ and $(-2, 0)$, and the distance from the center to each focus is $c = 2$. This result is in simplified and rationalized form, as required.

Additional Information

• The equation of an ellipse in standard form is given by $\frac{(x-h)^2}{a2} + \frac{(y-k)^2}{b2} = 1$, where $(h,k)$ is the center of the ellipse, and $a$ and $b$ are the semi-major and semi-minor axes, respectively.
• The foci of an ellipse are located on the major axis, which is the x-axis in this case.
• The distance between the center of the ellipse and each focus is given by the formula $c = \sqrt{a^2 - b^2}$, where $c$ is the distance from the center to each focus.

References

• [1] "Ellipses." Math Open Reference, mathopenref.com/ellipse.html.
• [2] "Ellipses." Khan Academy, khanacademy.org/math/geometry/ellipses.

Related Topics

• [1] "What is an Ellipse?" Math Is Fun, mathisfun.com/geometry/ellipse.html.
• [2] "Ellipses." Wikipedia, en.wikipedia.org/wiki/Ellipse.
  

Introduction

Ellipses are a fundamental concept in mathematics, and they have numerous applications in various fields such as physics, engineering, and computer science. In this article, we will address some of the most frequently asked questions about ellipses, providing detailed explanations and examples to help you better understand this concept.

Q1: What is an Ellipse?

A1: An ellipse 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. It is a two-dimensional shape that is symmetrical about its major and minor axes.

Q2: What is the Difference between an Ellipse and a Circle?

A2: A circle is a special type of ellipse where the major and minor axes are equal, and the distance between the center and each focus is zero. In other words, a circle is an ellipse with a zero eccentricity.

Q3: How to Find the Foci of an Ellipse?

A3: To find the foci of an ellipse, you need to rewrite the equation of the ellipse in standard form. The distance between the center of the ellipse and each focus is given by the formula $c = \sqrt{a^2 - b^2}$, where $c$ is the distance from the center to each focus.

Q4: What is the Eccentricity of an Ellipse?

A4: The eccentricity of an ellipse is a measure of how elliptical it is. It is defined as the ratio of the distance between the center and each focus to the length of the semi-major axis. The formula for eccentricity is $e = \frac{c}{a}$, where $e$ is the eccentricity, $c$ is the distance from the center to each focus, and $a$ is the length of the semi-major axis.

Q5: How to Find the Area of an Ellipse?

A5: The area of an ellipse is given by the formula $A = \pi ab$, where $A$ is the area, $a$ is the length of the semi-major axis, and $b$ is the length of the semi-minor axis.

Q6: What is the Perimeter of an Ellipse?

A6: The perimeter of an ellipse is a more complex concept, and there is no simple formula to calculate it. However, you can use numerical methods or approximations to find the perimeter of an ellipse.

Q7: How to Find the Equation of an Ellipse in Standard Form?

A7: To find the equation of an ellipse in standard form, you need to rewrite the given equation in the form $\frac{(x-h)^2}{a2} + \frac{(y-k)^2}{b2} = 1$, 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.

Q8: What is the Relationship between Ellipses and Circles?

A8: Ellipses and circles are related in that a circle is a special type of ellipse with a zero eccentricity. However, ellipses can have any value of eccentricity, ranging from 0 (a circle) to 1 (a parabola).

Q9: How to Graph an Ellipse?

A9: To graph an ellipse, you need to find the center, semi-major axis, and semi-minor axis of the ellipse. Then, you can plot the major and minor axes and use them to draw the ellipse.

Q10: What are the Applications of Ellipses?

A10: Ellipses have numerous applications in various fields such as physics, engineering, and computer science. Some examples include the orbits of planets, the shape of mirrors and lenses, and the design of electronic circuits.

Conclusion

In conclusion, we have addressed some of the most frequently asked questions about ellipses, providing detailed explanations and examples to help you better understand this concept. Whether you are a student, a teacher, or a professional, we hope that this article has been helpful in your understanding of ellipses.

Additional Information

• [1] "Ellipses." Math Open Reference, mathopenref.com/ellipse.html.
• [2] "Ellipses." Khan Academy, khanacademy.org/math/geometry/ellipses.
• [3] "Eccentricity of an Ellipse." Wikipedia, en.wikipedia.org/wiki/Eccentricity_(ellipse).

Related Topics

• [1] "What is an Ellipse?" Math Is Fun, mathisfun.com/geometry/ellipse.html.
• [2] "Ellipses." Wikipedia, en.wikipedia.org/wiki/Ellipse.
• [3] "Geometry." Khan Academy, khanacademy.org/math/geometry.