A) a circle
B) a parabola
C) an ellipse
D) a hyperbola
Correct Answer: C
Solution :
Let (h, k) be the mid point of a chord passing through the positive end of the minor axis of the ellipse \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1.\]Then, the equation of the chord is \[\frac{hx}{{{a}^{2}}}+\frac{ky}{{{b}^{2}}}-1=\frac{{{h}^{2}}}{{{a}^{2}}}+\frac{{{k}^{2}}}{{{b}^{2}}}-1\](using T = S?) \[\Rightarrow \]\[\frac{hx}{{{a}^{2}}}+\frac{ky}{{{b}^{2}}}=\frac{{{h}^{2}}}{{{a}^{2}}}+\frac{{{k}^{2}}}{{{b}^{2}}}\] This passes through (0, b). \[\therefore \]\[\frac{k}{b}=\frac{{{h}^{2}}}{{{a}^{2}}}+\frac{{{k}^{2}}}{{{b}^{2}}}\]. Hence, the locus of (h, k) is \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=\frac{y}{b},\] which represents an ellipse.
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