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Question

The curves satisfying the differential equation (1x2)y1+xy=ax are

A
ellipses and hyperbolas
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B
ellipses and parabola
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C
ellipses and straight lines
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D
circles and ellipses
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Solution

The correct option is A ellipses and hyperbolas
The given equation is linear DE and can be written as
dydx+x1x2y=ax1x2
Its integrating factor is ex1x2dx=e(12)log(1x2)=11x21<x<1 and ifx2>1 then I.F.=1x21
ddx(y11x2)=ax(1x2)32=12a2x(1x2)32
y11x2=a1x2+Cy=a+C1x2
(ya)2=C2(1x2)(ya)2+C2x2=C2
Thus if -1 < x < 1 the given equation represents an ellipse. If x2>1 then the solution is of the form (ya)2+C2x2=C2 which represents a hyperbola.

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