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Question

# Find the equation to the director circle of the hyperbola x2a2−y2b2=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=a2−b2 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 ⇒(x2a2−y2b2−1)(h2a2−k2−b2−1)=(hxa2−kyb2)2 x2a2(h2a2−k2b2−1)−y2b2(h2a2−k2b2−1)−(h2a2−k2b2−1) =h2x2a4+k2yb4+1−2hkxya2b2+2kyb2+2hxa2 x2a2(−k2b2−1)−y2b2(h2a2−1)−(h2a2−k2b2)+−2hkxya2b2−2kyb2+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(−k2b2−1)−1b2(h2a2−1)=0 −k2a2b2−1a2−h2a2b2+1b2=0

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