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

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SUMMARY

The length of the major axis of the ellipse with foci at (-41, 23) and (115, 42) is approximately 156.085 units. This value is derived using the distance formula, which calculates the distance between the two foci. Given that the ellipse is tangent to the x-axis, this distance directly corresponds to the length of the major axis. Thus, the major axis serves as the longest diameter connecting the vertices of the ellipse.

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  • Understanding of the distance formula in coordinate geometry
  • Knowledge of the properties of ellipses, including foci and major axis
  • Familiarity with the concept of tangency in geometry
  • Basic skills in algebra for manipulating equations
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  • Study the properties of ellipses, focusing on their geometric definitions
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  • Explore applications of the distance formula in different geometric contexts
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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?
 
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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.
 

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