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Question

The locus of a point, from where pair of tangents to the rectangular hyperbola x2y2=a2 contain an angle of 45, is :

A
(x2+y2)2+4a2(x2y2)=4a4
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B
(x2+y2)2+4a2(x2y2)=a4
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C
2(x2+y2)+4a2(x2y2)=4a2
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D
(x2+y2)+a2(x2y2)=4a2
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Solution

The correct option is A (x2+y2)2+4a2(x2y2)=4a4
Equation of tangent to given hyperbola is : y=mx±a2m2a2
Let P(h,k) be the locus
kmh=±a2m2a2
k2+m2h22kmh=a2m2a2
m2(h2a2)2hkm+k2+a2=0
m1+m2=2hkh2a2 and
m1m2=k2+a2h2a2
Now,
tan45=m1m21+m1m2
(m1m2)2=(1+m1m2)2
(m1+m2)24m1m2=(1+m1m2)2
(2hkh2a2)24(k2+a2h2a2)=(h2+k2h2a2)2
(h2+k2)2+4a2(h2k2)=4a4

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