Explanation
Step 1: Understand the Ellipse Orientation
For the given equation:
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If a > b, the major axis is along the x-axis.
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Since the question specifies a < b, the major axis is along the y-axis.
Step 2: Focal Distance Property
By the definition of an ellipse, the sum of the distances of any point P(x,y) on the ellipse from its two foci (S and S′) is always constant and equal to the length of the major axis.
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In a horizontal ellipse (a > b), the major axis length is 2a.
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In a vertical ellipse (a < b), the major axis length is 2b.
Step 3: Verification
The coordinates of the foci for a vertical ellipse (a < b) are:
For any point P on the ellipse, the sum of focal distances is:
PS+PS′=Length of Major Axis
Final Answer: The sum of the focal distances is 2b.
