An ellipse has its center at the origin, a focus at (3, 0), and a vertex at (5, 0). What is the length of the minor axis?
Show the answer and explanation
Correct answer
C. 8
Principle or equation
For an ellipse centered at the origin with major axis along the x-axis, the standard equation is x^2/a^2 + y^2/b^2 = 1, where a > b > 0. The distance from the center to a focus is c, with c^2 = a^2 - b^2. The length of the minor axis is 2b.
Why this answer is correct
Here the vertex at (5,0) gives a = 5. The focus at (3,0) gives c = 3. Then b^2 = a^2 - c^2 = 25 - 9 = 16, so b = 4. The minor axis length is 2b = 8.
Example
If a = 4 and c = 2, then b^2 = 16 - 4 = 12, so b = 2√3, minor axis = 4√3.
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