Find the equation of the ellipse with a major axis length of , a minor axis length of , center at , and foci on .

Question

Find the equation of the ellipse with a major axis length of  , a minor axis length of  , center at  , and foci on  .

Quizplus · Accepted Answer

The equation of an ellipse with center (h,k), major axis length 2a, and minor axis length 2b is $\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1$.

In this case, the center is (1,2), the major axis length is 11, and the minor axis length is 7. Using the formula $c^2 = a^2 - b^2$, we can find the distance from the center to each focus:

$c = \sqrt{a^2 - b^2} = \sqrt{11^2 - 7^2} = 4\sqrt{3}$

The foci are 4√3 units away from the center horizontally, so their x-coordinates are 1-4√3 and 1+4√3.

Substituting the known values into the equation, we get:

$$\frac{(x-1)^2}{11^2}+\frac{(y-2)^2}{7^2}=1$$

Simplifying gives the equation:

$$\frac{(x-1)^2}{121}+\frac{(y-2)^2}{49}=1$$

Therefore, the answer is B.

Find the Equation of the Ellipse with a Major Axis