A) \[\frac{{{x}^{2}}}{{{k}^{2}}}+\frac{{{y}^{2}}}{{{k}^{2}}+{{h}^{2}}}=1\]
B) \[\frac{{{x}^{2}}}{{{k}^{2}}}+\frac{{{y}^{2}}}{{{h}^{2}}-{{k}^{2}}}=1\]
C) \[\frac{{{x}^{2}}}{{{k}^{2}}}+\frac{{{y}^{2}}}{{{k}^{2}}-{{h}^{2}}}=1\]
D) \[\frac{{{x}^{2}}}{{{k}^{2}}}+\frac{{{y}^{2}}}{{{h}^{2}}}=1\]
Correct Answer: C
Solution :
[c] Let the equation of the ellipse. \[\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1\] Let e be the eccentricity of the ellipse. Since distance between foci = 2h \[\therefore \,\,\,\,\,2ae=2h\Rightarrow e=h\] ?. (1) Focal distance of one end of minor axis say (0, b) is k \[\therefore a+e(0)=k\Rightarrow a=k\] ?. (2) From (1) and (2), \[{{b}^{2}}={{a}^{2}}(1-{{e}^{2}})={{k}^{2}}-{{h}^{2}}\] \[\therefore \] The equation of the ellipse is \[\frac{{{x}^{2}}}{{{k}^{2}}}+\frac{{{y}^{2}}}{{{k}^{2}}-{{h}^{2}}}=1.\]
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