JEE Main & Advanced
Sample Paper
JEE Main Sample Paper-24
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 \[\left( \pm \,\frac{2}{\alpha }\,\sqrt{1+{{\alpha }^{2}}}\,,\,\,0 \right)\] \[\therefore \,\,\frac{4}{{{\alpha }^{2}}}\,(1+{{\alpha }^{2}})=5\Rightarrow \,\frac{4}{{{\alpha }^{2}}}\,+4=5\]\[\Rightarrow \,{{\alpha }^{2}}=4\Rightarrow \,\alpha =2\]