Understanding the Properties of Ellipses: From Covertices to Foci

Ellipses are fascinating geometric shapes with a variety of properties and applications. In this article, we'll delve into the specifics of an ellipse with covertices at (5, 6) and (5, 8), and where the sum of the distances from any point on the ellipse to the foci is 12.

Introduction to Ellipses and Given Conditions

Ellipses are conic sections characterized by their symmetrical properties. The covertices of an ellipse are the endpoints of the minor axis, and the foci are points inside the ellipse that help define its shape and properties.

Given Data:

Covertices: (5, 6) and (5, 8) Sum of distances from any point on the ellipse to the foci: 12

Steps to Determine the Standard Equation of the Ellipse

1. Identifying the Covertices and the Ellipse Orientation

The covertices, (5, 6) and (5, 8), share the same x-coordinate, which means the minor axis is vertical. Therefore, the major axis is horizontal, and the entire ellipse is vertical.

2. Calculating the Center of the Ellipse

The center of the ellipse is the midpoint of the covertices. To find this, we use the midpoint formula:

[ text{Center} left(frac{5 5}{2}, frac{6 8}{2}right) (5, 7) ]

3. Calculating the Minor Axis Length (2b)

The length of the minor axis is the difference in the y-coordinates of the covertices:

[ 8 - 6 2 ]

Therefore, the semi-minor axis (b) is:

[ b frac{2}{2} 1 ]

4. Calculating the Distance from the Center to the Foci (2a)

The sum of the distances from any point on the ellipse to the foci is given as 12. For an ellipse, this sum is equal to 2a. Therefore:

[ 2a 12 text{ implies } a 6 ] [ a^2 36 ]

5. Calculating the Distance from the Center to the Foci (c)

The relationship between the semi-major axis (a), semi-minor axis (b), and the distance from the center to the foci (c) is given by:

[ c^2 a^2 - b^2 ] [ c^2 36 - 1^2 35 ] [ c sqrt{35} ]

Data Points and Key Calculations

Center: (5, 7)

a (semi-major axis): 6

b (semi-minor axis): 1

c (distance from the center to the foci): ( sqrt{35} )

The Standard Equation of the Ellipse

The standard form of the equation for a vertical ellipse centered at (h, k) is:

[ frac{(x-h)^2}{b^2} frac{(y-k)^2}{a^2} 1 ]

Substituting the known values (h 5, k 7, a 6, and b 1) into the standard equation:

[ frac{(x-5)^2}{1^2} frac{(y-7)^2}{6^2} 1 ]

This simplifies to:

[ frac{(x-5)^2}{1} frac{(y-7)^2}{36} 1 ]

Therefore, the standard equation of the ellipse is:

[ frac{(x-5)^2}{1} frac{(y-7)^2}{36} 1 ]

Conclusion

In conclusion, using the properties of covertices, the center of the ellipse, the minor axis length, and the distance from the center to the foci, we were able to determine the standard equation of the ellipse. This method can be applied to similar problems, making it a useful skill in understanding and manipulating ellipses.

Related Keywords

Ellipse

Covertices

Foci

Standard Equation

References

Calculus Book - Calculating the Properties of Ellipses

Geometry Textbook - Ellipse Equations and Properties

Online Calculus Resources - Ellipse Calculations