What Is The Length Of The Major Axis Of The Ellipse Given By The Equation $\frac{x^2}{10} + \frac{y^2}{20} = 1$?

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 $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$, where $a$ and $b$ are the lengths of the semi-major and semi-minor axes, respectively. In this article, we will discuss how to find the length of the major axis of an ellipse given by the equation $\frac{x^2}{10} + \frac{y^2}{20} = 1$.

Understanding the Equation of the Ellipse

The given equation of the ellipse is $\frac{x^2}{10} + \frac{y^2}{20} = 1$. To find the length of the major axis, we need to identify the values of $a$ and $b$ in the equation. Comparing the given equation with the standard form of the equation of an ellipse, we can see that $a^2 = 10$ and $b^2 = 20$. Taking the square root of both sides, we get $a = \sqrt{10}$ and $b = \sqrt{20}$.

Finding the Length of the Major Axis

The length of the major axis of an ellipse is given by $2a$, where $a$ is the length of the semi-major axis. Since we have found the value of $a$ to be $\sqrt{10}$, we can now find the length of the major axis by multiplying $a$ by 2. Therefore, the length of the major axis of the ellipse given by the equation $\frac{x^2}{10} + \frac{y^2}{20} = 1$ is $2\sqrt{10}$.

Simplifying the Answer

To simplify the answer, we can rationalize the denominator by multiplying the numerator and denominator by $\sqrt{10}$. This gives us $\frac{2\sqrt{10}\sqrt{10}}{\sqrt{10}} = \frac{20}{\sqrt{10}}$. To further simplify the answer, we can multiply the numerator and denominator by $\sqrt{10}$ again. This gives us $\frac{20\sqrt{10}}{\sqrt{10}\sqrt{10}} = \frac{20\sqrt{10}}{10} = 2\sqrt{10}$.

Conclusion

In conclusion, the length of the major axis of the ellipse given by the equation $\frac{x^2}{10} + \frac{y^2}{20} = 1$ is $2\sqrt{10}$. This can be simplified to $2\sqrt{10}$.

Final Answer

The final answer is: $\boxed{2\sqrt{10}}$

Additional Information

• The length of the semi-major axis is $\sqrt{10}$.
• The length of the semi-minor axis is $\sqrt{20}$.
• The equation of the ellipse can be written as $\frac{x^2}{(\sqrt{10})^2} + \frac{y^2}{(\sqrt{20})^2} = 1$.

References

• [1] "Ellipses" by Math Open Reference. Math Open Reference. https://www.mathopenref.com/ellipse.html
• [2] "Equation of an Ellipse" by Purplemath. Purplemath. https://www.purplemath.com/modules/ellipse.htm

Related Topics

• Equation of an Ellipse
• Length of the Semi-Major Axis
• Length of the Semi-Minor Axis

Tags

• Ellipse
• Equation of an Ellipse
• Length of the Major Axis
• Length of the Semi-Major Axis
• Length of the Semi-Minor Axis
  

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

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

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

A: To find the length of the major axis of an ellipse, you need to identify the values of $a$ and $b$ in the equation of the ellipse. The length of the major axis is given by $2a$, where $a$ is the length of the semi-major axis.

Q: What is the length of the semi-major axis of the ellipse given by the equation $\frac{x^2}{10} + \frac{y^2}{20} = 1$?

A: The length of the semi-major axis of the ellipse given by the equation $\frac{x^2}{10} + \frac{y^2}{20} = 1$ is $\sqrt{10}$.

Q: What is the length of the semi-minor axis of the ellipse given by the equation $\frac{x^2}{10} + \frac{y^2}{20} = 1$?

A: The length of the semi-minor axis of the ellipse given by the equation $\frac{x^2}{10} + \frac{y^2}{20} = 1$ is $\sqrt{20}$.

Q: How do I simplify the answer to find the length of the major axis?

A: To simplify the answer, you can rationalize the denominator by multiplying the numerator and denominator by $\sqrt{10}$. This gives you $\frac{2\sqrt{10}\sqrt{10}}{\sqrt{10}} = \frac{20}{\sqrt{10}}$. To further simplify the answer, you can multiply the numerator and denominator by $\sqrt{10}$ again. This gives you $\frac{20\sqrt{10}}{\sqrt{10}\sqrt{10}} = \frac{20\sqrt{10}}{10} = 2\sqrt{10}$.

Q: What is the final answer to the problem?

A: The final answer to the problem is $2\sqrt{10}$.

Q: What are some additional information related to the problem?

A: Some additional information related to the problem includes:

• The length of the semi-major axis is $\sqrt{10}$.
• The length of the semi-minor axis is $\sqrt{20}$.
• The equation of the ellipse can be written as $\frac{x^2}{(\sqrt{10})^2} + \frac{y^2}{(\sqrt{20})^2} = 1$.

Q: What are some related topics to the problem?

A: Some related topics to the problem include:

• Equation of an Ellipse
• Length of the Semi-Major Axis
• Length of the Semi-Minor Axis

Q: What are some references related to the problem?

A: Some references related to the problem include:

• [1] "Ellipses" by Math Open Reference. Math Open Reference. https://www.mathopenref.com/ellipse.html
• [2] "Equation of an Ellipse" by Purplemath. Purplemath. https://www.purplemath.com/modules/ellipse.htm

Q: What are some tags related to the problem?

A: Some tags related to the problem include:

• Ellipse
• Equation of an Ellipse
• Length of the Major Axis
• Length of the Semi-Major Axis
• Length of the Semi-Minor Axis