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Q.

The locus of the point of intersection of tangents to the hyperbola x2 y2 =a2 which include an angle of 45º is

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a

(x2 +y2)2 = 4a2(x2+y2+a2)

b

(x2 +y2)2 = 4a2(x2-y2+a2)

c

(x2 +y2)2 = 4a2(y2-x2+a2)

d

(x2 +y2)2 = 4a2(x2+y2-a2)

answer is C.

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Detailed Solution

Let y=mx±a2m2-a2 be two tangents to the hyperbola. If it passes through (x1, y1) then (y1-mx1)2=a2m2-a2 m2(x12-a2)-2mx1y1+y12+a2=0 m1+m2=2x1y1x12-a2, m1m2=y12+a2x12-a2 If 'θ' is angle between tangents then  tan45=m1-m21+m1m2 1=m1-m21+m1m2 m1-m2=1+m1m2 (m1-m2)2=(1+m1m2)2  (m1+m2)2-4m1m2=(1+m1m2)2 (2x1y1x12-a2)2-4(y12+a2x12-a2)=(1+y12+a2x12-a2)2  4x12y12-4(y12+a2)(x12-a2)=(x12-a2+y12+a2)2  4a2(y12-x12+a2)=(x21+y12)2 locus is (x2 +y 2)2=4a2(y2-x2+a2)

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The locus of the point of intersection of tangents to the hyperbola x2 –y2 =a2 which include an angle of 45º is