Problem 30 - College Algebra

Question

Find the standard form of the equation of each ellipse satisfying the given conditions. Foci: \((0,-2),(0,2) ; x\) -intercepts: \(-2\) and 2

Step-by-Step Solution

Answer

So, The standard form of the equation of the ellipse is \( \frac{x^2}{4} + \frac{y^2}{4} = 1\).

Step 1: Determine the Orientation of the Ellipse

Looking at the coordinates of foci \((0, -2), (0, 2)\), one can see they're located along the y-axis. This means our ellipse is oriented vertically. The standard form equation is \( \frac{x^2}{b^2} + \frac{y^2}{a^2} = 1\) for a vertically oriented ellipse where \(a\) is the distance from center to vertex along the y-axis, and \(b\) is the distance from center to vertex along the x-axis.

Step 2: Calculate the Values of a and b

Given the foci, we have the value of \(a\) which is the distance from the center of the ellipse (origin in our case) to the foci. So, the distance between the origin and either of the foci gives \(a = 2\). In addition, the x-intercepts are given as -2 and 2. Therefore, the distance from the center to either x-intercept gives the value of \(b = 2\). Using these, we can build our ellipse equation.

Step 3: Substitute a and b into the Equation of an Ellipse

The values of \(a\) and \(b\) obtained from step 2 are substituted into the standard equation of the ellipse. So, our equation becomes \( \frac{x^2}{4} + \frac{y^2}{4} = 1\).