Problem 27 - College Algebra

Question

Find the standard form of the equation of each ellipse satisfying the given conditions. $$\text { Foci: }(0,-4),(0,4) ; \text { vertices: }(0,-7),(0,7)$$

Step-by-Step Solution

Answer

The standard form of the equation of the ellipse is \(x^2/33 + y^2/49 = 1\)

Step 1: Determine the values of a and c

The value of a (distance from the center to the vertices) can be obtained from the coordinates of the vertices, which is the y-coordinate of vertex as origin is at the center. So \(a = 7\). The value of c (distance from the center to the foci) can be obtained from the coordinates of the foci, which is the y-coordinate of focus as origin is at the center. So \(c = 4\).

Step 2: Find the value of b

Substitute the values of a and c into the relationship \(a^2 = b^2 + c^2\) to get: \((7)^2 = b^2 + (4)^2\), which simplifies to \(49 = b^2 + 16\). Solving this for b^2 gives, \(b^2 = 49 - 16 = 33\).

Step 3: Write the standard form of the equation of the ellipse

Now you have a, b, and c. Substitute these in the standard form equation for a vertical ellipse, \(x^2/b^2 + y^2/a^2 = 1\), and you get the equation of the ellipse: \(x^2/33 + y^2/49 = 1\).