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Question

Find the equation to the director circle of the hyperbola x2a2y2b2=1


A

x2+y2=a2b2

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B

x2y2=a2b2

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C

x2+y2=a2+b2

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D

x2y2=a2+b2

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Solution

The correct option is A

x2+y2=a2b2


Director circle is the locus of the point of intersection of the tangents which are right angled.

Let P(h,k) be the point of intersection of two perpendicular tangents.

Equation of pair of tangents is ss1=T2

(x2a2y2b21)(h2a2k2b21)=(hxa2kyb2)2

x2a2(h2a2k2b21)y2b2(h2a2k2b21)(h2a2k2b21)

=h2x2a4+k2yb4+12hkxya2b2+2kyb2+2hxa2

x2a2(k2b21)y2b2(h2a21)(h2a2k2b2)+2hkxya2b22kyb2+2hxa2=0 ........(1)

Comparing this equation with standard pair of line eqution

ax2+by2+2hxy+2gx+2fy+c=0

Since, equation (1) represents two perpendicular line

a+b=0 or

Coefficient of x2+ coefficient of y2=0

1a2(k2b21)1b2(h2a21)=0

k2a2b21a2h2a2b2+1b2=0


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