What is the length of the major axis of the ellipse?

[Greg Bernhardt]

An ellipse on the xy-plane has foci at (-41, 23) and (115, 42). The ellipse is tangent to the x-axis. What is the length of the major axis of the ellipse?

 

[Greg Bernhardt]

point for teddy!

 

[R0man]

The length of the major axis of an ellipse is the longest diameter that runs through the center of the ellipse and connects two opposite points on the ellipse called the vertices. In this case, the foci and the tangent point on the x-axis can be used to determine the length of the major axis.

Using the distance formula, we can calculate the distance between the two foci as follows:

d = √[(x2 - x1)^2 + (y2 - y1)^2]
= √[(115 - (-41))^2 + (42 - 23)^2]
= √[156^2 + 19^2]
= √24337
= 156.085

Since the ellipse is tangent to the x-axis, the distance between the foci is equal to the length of the major axis. Therefore, the length of the major axis of the ellipse is approximately 156.085 units.