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Question

If the line x+2y+4=0 cutting the ellipse in points whose eccentric angles are 30 and 60 subtends a right angle at the origin, then its equation is

A
x24+y216=1
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B
x216+y24=1
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C
x28+y24=1
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D
x24+y28=1
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Solution

The correct option is B x216+y24=1
Let P(acos30,bsin30) and Q be (acos60,bsin60)
Its slope =b(sin60sin30)a(cos60cos30)=ba
But slope of x+2y+4=0 is 12 ba=12 or a=2b
The chord subtends an angle of 90 at the origin.
Making equation of the ellipse homogeneous with the equation of line, we have x2a2+y2b2=(x+2y4)2 or x2(1a2116)xy4+y2(1b214)=0
Since the line subtends an angle of 90 at origin A+B=0 or 1a2116+1b214=0 or 14b2116+1b214=0 a=2b,by(1) 54b2=516 b2=4a2=4b2=16
Hence the equation is x216+y24=1

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