Ellipse

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Question

An ellipse passing through the origin has its foci (3, 4) and (6, 8). The length of its semi-minor axis is 6. Then the value of $b / \sqrt{2}$ is _____.

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Solution

The points are A(3,4),B(6, 8), and O(0, 0).
OA + OB = 2a (where a is semi-major axis)
$2 a = 5 + 10 = 15 ∴ a = \frac{15}{2} Now, 2 ae = \sqrt{(6 - 3)^{2} + (8 - 4)^{2}} = 5 \begin{array}{ll} ∴ e = \frac{1}{3} \\ ∴ b^{2} = \frac{225}{4} (1 - \frac{1}{9}) = 50 \end{array}$

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