Find The Ends Of The Major Axis And Foci Of The Ellipse:$\[\frac{x^2}{144} + \frac{y^2}{169} = 1\\]Major Axis: \[$(0, \pm[?])\$\] Foci: \[$(0, \pm[\quad])\$\]

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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 major axis of an ellipse is the longest diameter that can be drawn through the ellipse, passing through both foci. In this article, we will explore how to find the ends of the major axis and the foci of an ellipse given its equation.

The Equation of an Ellipse

The general equation of an ellipse centered at the origin (0, 0) is given by:

x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

where aa and bb are the lengths of the semi-major and semi-minor axes, respectively.

The Given Equation

The given equation of the ellipse is:

x2144+y2169=1\frac{x^2}{144} + \frac{y^2}{169} = 1

We can see that a2=144a^2 = 144 and b2=169b^2 = 169. Therefore, a=12a = 12 and b=13b = 13.

Finding the Ends of the Major Axis

The major axis of an ellipse is the longest diameter that can be drawn through the ellipse, passing through both foci. Since the ellipse is centered at the origin, the major axis lies along the y-axis. The ends of the major axis are given by:

(0,±a)(0, \pm a)

Substituting the value of aa, we get:

(0,±12)(0, \pm 12)

Therefore, the ends of the major axis are (0,12)(0, 12) and (0,−12)(0, -12).

Finding the Foci

The foci of an ellipse are the two points inside the ellipse that are equidistant from the center. The distance between the center and each focus is given by:

c=a2−b2c = \sqrt{a^2 - b^2}

Substituting the values of aa and bb, we get:

c=144−169=−25=5ic = \sqrt{144 - 169} = \sqrt{-25} = 5i

However, since cc is a real number, we can rewrite it as:

c=25=5c = \sqrt{25} = 5

The foci are located at a distance of cc from the center along the y-axis. Therefore, the foci are given by:

(0,±c)(0, \pm c)

Substituting the value of cc, we get:

(0,±5)(0, \pm 5)

Therefore, the foci are (0,5)(0, 5) and (0,−5)(0, -5).

Conclusion

In conclusion, we have found the ends of the major axis and the foci of the given ellipse. The ends of the major axis are (0,12)(0, 12) and (0,−12)(0, -12), and the foci are (0,5)(0, 5) and (0,−5)(0, -5). This demonstrates the importance of understanding the properties of an ellipse, including its major axis and foci.

Key Takeaways

  • The major axis of an ellipse is the longest diameter that can be drawn through the ellipse, passing through both foci.
  • The ends of the major axis are given by (0,±a)(0, \pm a).
  • The foci of an ellipse are the two points inside the ellipse that are equidistant from the center.
  • The distance between the center and each focus is given by c=a2−b2c = \sqrt{a^2 - b^2}.

Further Reading

For further reading on the topic of ellipses, we recommend the following resources:

References

  • [1] "Geometry: A Comprehensive Introduction" by Dan Pedoe
  • [2] "Calculus: Early Transcendentals" by James Stewart
  • [3] "Mathematics for Computer Science" by Eric Lehman, F Thomson Leighton, and Albert R Meyer
    Frequently Asked Questions (FAQs) about Ellipses =====================================================

Q: What is an ellipse?

A: 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.

Q: What is the major axis of an ellipse?

A: The major axis of an ellipse is the longest diameter that can be drawn through the ellipse, passing through both foci.

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

A: To find the ends of the major axis, you need to know the length of the semi-major axis, which is given by aa. The ends of the major axis are given by (0,±a)(0, \pm a).

Q: What are the foci of an ellipse?

A: The foci of an ellipse are the two points inside the ellipse that are equidistant from the center.

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

A: To find the foci, you need to know the length of the semi-major axis, aa, and the length of the semi-minor axis, bb. The distance between the center and each focus is given by c=a2−b2c = \sqrt{a^2 - b^2}.

Q: What is the equation of an ellipse?

A: The general equation of an ellipse centered at the origin (0, 0) is given by:

x2a2+y2b2=1\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1

Q: How do I determine the values of aa and bb in the equation of an ellipse?

A: To determine the values of aa and bb, you need to know the lengths of the semi-major and semi-minor axes, respectively. These values can be obtained from the equation of the ellipse.

Q: What is the significance of the foci of an ellipse?

A: The foci of an ellipse are significant because they help to define the shape and size of the ellipse. The distance between the center and each focus is given by c=a2−b2c = \sqrt{a^2 - b^2}.

Q: Can an ellipse have more than two foci?

A: No, an ellipse can only have two foci.

Q: Can an ellipse be a circle?

A: Yes, an ellipse can be a circle if the lengths of the semi-major and semi-minor axes are equal, i.e., a=ba = b.

Q: What is the relationship between an ellipse and a circle?

A: 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. A circle is a special case of an ellipse where the lengths of the semi-major and semi-minor axes are equal.

Q: Can an ellipse be a parabola?

A: No, an ellipse cannot be a parabola.

Q: What is the difference between an ellipse and a hyperbola?

A: 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. A hyperbola is a pair of open curves on a plane that are symmetric about the x-axis and y-axis.

Q: Can an ellipse be a hyperbola?

A: No, an ellipse cannot be a hyperbola.

Q: What is the significance of the major axis of an ellipse?

A: The major axis of an ellipse is significant because it helps to define the shape and size of the ellipse.

Q: Can the major axis of an ellipse be vertical or horizontal?

A: The major axis of an ellipse can be either vertical or horizontal, depending on the orientation of the ellipse.

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

A: The semi-major axis of an ellipse is significant because it helps to define the shape and size of the ellipse.

Q: Can the semi-major axis of an ellipse be negative?

A: No, the semi-major axis of an ellipse cannot be negative.

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

A: The semi-minor axis of an ellipse is significant because it helps to define the shape and size of the ellipse.

Q: Can the semi-minor axis of an ellipse be negative?

A: No, the semi-minor axis of an ellipse cannot be negative.

Q: What is the relationship between the semi-major axis and the semi-minor axis of an ellipse?

A: The semi-major axis and the semi-minor axis of an ellipse are related by the equation c=a2−b2c = \sqrt{a^2 - b^2}, where cc is the distance between the center and each focus.

Q: Can the semi-major axis and the semi-minor axis of an ellipse be equal?

A: Yes, the semi-major axis and the semi-minor axis of an ellipse can be equal, in which case the ellipse is a circle.

Q: What is the significance of the eccentricity of an ellipse?

A: The eccentricity of an ellipse is significant because it helps to define the shape and size of the ellipse.

Q: Can the eccentricity of an ellipse be negative?

A: No, the eccentricity of an ellipse cannot be negative.

Q: What is the relationship between the eccentricity and the semi-major axis of an ellipse?

A: The eccentricity of an ellipse is related to the semi-major axis by the equation e=cae = \frac{c}{a}, where cc is the distance between the center and each focus.

Q: Can the eccentricity of an ellipse be zero?

A: Yes, the eccentricity of an ellipse can be zero, in which case the ellipse is a circle.

Q: What is the significance of the focal parameter of an ellipse?

A: The focal parameter of an ellipse is significant because it helps to define the shape and size of the ellipse.

Q: Can the focal parameter of an ellipse be negative?

A: No, the focal parameter of an ellipse cannot be negative.

Q: What is the relationship between the focal parameter and the semi-major axis of an ellipse?

A: The focal parameter of an ellipse is related to the semi-major axis by the equation p=a2cp = \frac{a^2}{c}, where cc is the distance between the center and each focus.

Q: Can the focal parameter of an ellipse be zero?

A: Yes, the focal parameter of an ellipse can be zero, in which case the ellipse is a circle.