An ellipse has a center at the origin, a vertex along the major axis at $(10,0)$, and a focus at $(8,0)$.
Which equation represents this ellipse?
$\frac{x^2}{10^2}+\frac{y^2}{6^2}=1$
$\frac{x^2}{10^2}+\frac{y^2}{8^2}=1$
$\frac{y^2}{10^2}+\frac{x^2}{6^2}=1$
$\frac{y^2}{10^2}+\frac{x^2}{8^2}=1$
Answer (1)
The ellipse has its major axis along the x-axis, so its equation is of the form a 2 x 2 + b 2 y 2 = 1 .
The vertex at ( 10 , 0 ) gives a = 10 , and the focus at ( 8 , 0 ) gives c = 8 .
Using the relationship a 2 = b 2 + c 2 , we find b 2 = a 2 − c 2 = 1 0 2 − 8 2 = 100 − 64 = 36 , so b = 6 .
Substituting a = 10 and b = 6 into the equation, we get the equation of the ellipse: 1 0 2 x 2 + 6 2 y 2 = 1 .
Explanation
Analyze the given information and state the general form of the ellipse equation. The ellipse is centered at the origin, and the vertex and focus are on the x-axis. This tells us that the major axis is along the x-axis. The general form of an ellipse with its center at the origin and major axis along the x-axis is given by: a 2 x 2 + b 2 y 2 = 1 where a is the semi-major axis and b is the semi-minor axis.
Determine the values of a and c. The vertex is at ( 10 , 0 ) , which means the semi-major axis a = 10 . The focus is at ( 8 , 0 ) , which means the distance from the center to the focus is c = 8 . We know that for an ellipse, the relationship between a , b , and c is given by: a 2 = b 2 + c 2
Calculate the value of b. Substitute the values of a and c into the equation: 1 0 2 = b 2 + 8 2 100 = b 2 + 64 b 2 = 100 − 64 b 2 = 36 b = 6
Write the equation of the ellipse. Now we have a = 10 and b = 6 . Substitute these values into the general equation of the ellipse: 1 0 2 x 2 + 6 2 y 2 = 1 This matches the first option.
State the final answer. The equation that represents the ellipse is 1 0 2 x 2 + 6 2 y 2 = 1 .
Examples
Ellipses are commonly used in architecture and engineering. For example, the design of elliptical arches in bridges ensures optimal distribution of weight and stress. The properties of ellipses are also utilized in the construction of whispering galleries, where sound waves concentrate at the foci of the ellipse, allowing people at one focus to clearly hear whispers from the other focus. Understanding the equation and properties of ellipses helps engineers and architects create efficient and aesthetically pleasing structures.
