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Chapter 6: Problem 14

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertices: (0,卤8)\(;\) foci: (0,卤4)

Short Answer

Expert verified
The standard form of the given ellipse is \(x^2/48 + y^2/64 = 1\).

Step by step solution

01

Determine 'a'

Given the vertices are at (0,卤8), 'a' is equal to 8. This is because in an ellipse centered at the origin, 'a' is the distance from the center to the vertices. Thus, \(a = 8\).
02

Determine 'c'

Given the foci are at (0,卤4), 'c' is equal to 4. Here, 'c' is the distance from the center to the foci. Thus, \(c = 4\).
03

Determine 'b'

Using the formula \(a^2 = b^2 + c^2\), and substituting for 'a' and 'c', we get \(64 = b^2 + 16\). Solving this for 'b', we get \(b^2 = 48\) and thus \(b = \sqrt{48}, b \approx 6.93\).
04

Write the standard form of the ellipse

The standard form for a vertical ellipse centered at the origin is \(x^2/b^2 + y^2/a^2 = 1\). Substituting 'a' and 'b' from the previous steps, the equation of the ellipse becomes \(x^2/48 + y^2/64 = 1\).

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