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Chapter 6: Problem 23

Find the standard form of the equation of the ellipse with the given characteristics. Foci: (0,0),(0,8)\(;\) major axis of length 16

Short Answer

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

Step by step solution

01

Identify the semi major and semi minor axes

Given the length of the major axis, 16, the semi major axis 'a' is half of it, so a =16/2 = 8. Here, the foci are at (0,0) and (0,8), which indicates they lie along the y-axis, so the semi major axis lies along the y-axis. Also, the distance between the origin (which is also the ellipse's center) and either focus is the value 'c', which represents the focal length, so we get c = 4.
02

Calculate the semi minor axis length

The relationship between a, b, and c in an ellipse is given by \(a^2 = b^2 + c^2\). We know the values of a and c, so we can substitute them into this equation to solve for \(b^2\), the square of the semi minor axis length. Then b^2 = 64 - 16 = 48. The semi minor axis 'b' is then the square root of 48, i.e., \(b = \sqrt{48}\).
03

Write down the equation of the ellipse

The standard form of the equation of an ellipse with its center at the origin is \(x^2/b^2 + y^2/a^2 =1\). Here, the semi major axis lies along the y-axis, so the equation for this ellipse is \(x^2/48 + y^2/64 =1\).

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