Find the equation of the ellipse with foci and such that the distances from the foci to any point on the ellipse sum to 34.

Question

Find the equation of the ellipse with foci  and such that the distances from the foci to any point on the ellipse sum to 34.

Quizplus · Accepted Answer

Since the distances from the foci to any point on the ellipse sum to 34, we know that the major axis is 34. Additionally, we know that the distance between the foci is $2\sqrt{a^2-b^2}$. We can solve for $b$ using the Pythagorean theorem:

$(2\sqrt{a^2-b^2})^2 = (2a)^2 - 34^2$

$4a^2 - 4b^2 = 4a^2 - 1156$

$b^2 = \frac{1156}{4} = 289$

$b = 17$

Now we can use the standard form of an ellipse:

$\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

where $(h,k)$ is the center of the ellipse. Since the foci are at $(\pm c, 0)$, we know that the center is at $(0,0)$.

Plugging in our values of $a$ and $b$, we get:

$\frac{x^2}{a^2} + \frac{y^2}{289} = 1$

We just need to solve for $a$. We know that $2a$ is the length of the major axis, which we already found to be 34. Therefore, $a=17$.

The equation of the ellipse is:

$\frac{x^2}{289} + \frac{y^2}{289} = 1$

which simplifies to:

$x^2 + y^2 = 289$

Therefore, the answer is (C).

Find the Equation of the Ellipse with Foci and Such