With the origin as a focus and x = 4 as the c | vedclass

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(B) Let the focus be $F(0, 0)$ and the directrix be $x = 4$. Let the end of the minor axis be $B(h, k)$.By the definition of an ellipse, the distance

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a parabola

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(B) Let the focus be $F(0, 0)$ and the directrix be $x = 4$. Let the end of the minor axis be $B(h, k)$.By the definition of an ellipse, the distance from the focus to the point $B$ is $e$ times the distance from $B$ to the directrix.$BF = e \cdot BM$$BF = \sqrt{h^2 + k^2}$ and $BM = |4 - h|$.So, $\sqrt{h^2 + k^2} = e(4 - h)$.Also, for an ellipse, the distance from the center to the focus is $ae$ and the distance from the center to the directrix is $a/e$. Since the focus is at $(0,0)$ and the directrix is $x=4$, the distance between them is $a/e - ae = 4$.For the end of the minor axis, the distance from the center to the focus is $ae$, so $h = -ae$ and $k = b$.From $a/e - ae = 4$, we have $a(1 - e^2) = 4e$, which implies $b^2/a = 4e$, so $b^2 = 4ae$.Substituting $h = -ae$ and $k = b$, we get $k^2 = 4(-h) = -4h$.Replacing $(h, k)$ with $(x, y)$, the locus is $y^2 = -4x$, which is a parabola.

With the origin as a focus and $x = 4$ as the corresponding directrix, a family of ellipses is drawn. Then the locus of an end of the minor axis is

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