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Question

If the ellipse 4x2+9y2=36 and the hyperbola a2x2−y2=4 intersect at right angles, then which among the following options are correct

A
a=2
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B
a=4
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C
Equation of circle through the points of intersection of two conic is x2+y2=5
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D
Equation of circle through the points of intersection of two conic is x2+y2=25
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Solution

The correct options are
A a=2
C Equation of circle through the points of intersection of two conic is x2+y2=5
If ellipse and hyperbola are having same centre and intersecting each other orthogonally, then thay will be confocal
2ae=52a1+44/a2=5a=±2
Now equation of the curve which passes through the intersection of
E:4x2+9y2=36 and
H:4x2y2=4 is
E+λH=04x2(1+λ)+y2(9λ)364λ=0
For the curve to represent the circle
4(1+λ)=9λλ=1
Hence equation of the required circle
x2+y2=5

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