question_answer

If the ellipse \[4{{x}^{2}}+9{{y}^{2}}=36\] and the hyperbola \[{{\alpha }^{2}}{{x}^{2}}-{{y}^{2}}=4\] intersects orthogonally, then the value of \[\alpha \] can be

A)  5                                

B)  4

C)  3                                

D)  2

Correct Answer: D

Solution :

Since the ellipse and hyperbola intersect orthogonally then they are confocal. For ellipse \[\frac{{{x}^{2}}}{9}+\frac{{{y}^{2}}}{4}=1,\] the foci are\[\left( \pm \,\sqrt{5},\,0 \right)\] and for hyperbola \[\frac{{{x}^{2}}}{9}\,+\frac{{{y}^{2}}}{4}=1,\] foci are \[\left( \pm \,\frac{2}{\alpha }\,\sqrt{1+{{\alpha }^{2}}}\,,\,\,0 \right)\] \[\therefore \,\,\frac{4}{{{\alpha }^{2}}}\,(1+{{\alpha }^{2}})=5\Rightarrow \,\frac{4}{{{\alpha }^{2}}}\,+4=5\]\[\Rightarrow \,{{\alpha }^{2}}=4\Rightarrow \,\alpha =2\]