Question

Which is the equation of an ellipse with directrices at x = 14 and foci at (2.0) and (-2, 0)?

Answer 1

Answer:

  $\dfrac{x^2}{8}+\dfrac{y^2}{4}=1$

Step-by-step explanation:

The relationship between the focus, directrix, and semi-major axis is ...

  foci are ±ae

  directrices are ±a/e

where "a" is the length of the semi-major axis and e is the eccentricity.

Using the given focus and directrix locations, we have ...

  ae = 2

  a/e = 4

When we multiply these equations, we have ...

  (ae)(a/e) = (2)(4)

  a^2 = 8

This is the denominator of the x^2 term in the equation for an ellipse:

  x^2/a^2 + y^2/b^2 = 1

_____

We can find b^2 from ...

  ae = √(a^2 -b^2)

  b^2 = (a^2 -(ae)^2) = 8 -2^2 = 4

So, our ellipse equation is ...

  x^2/8 +y^2/4 = 1 . . . . . . matches the first choice