If the lengths of the tangents from two points A, B to a circle are 6, 7 respect

Question

If the lengths of the tangents from two points A, B to a circle are 6, 7 respectively. If A, B are conjugate points then AB =

Options:

A . 5

B . √85

C . √852

D . none

Answer: Option B
:
B
Let the equation of the circle be

x^{2}

y^{2} = a^{2}

and A

(x_{1}, y_{1})

B (x_{2}, y_{2})

A, B are conjugate points

\Rightarrow x_{1} x_{2} + y_{1} y_{2} = a^{2}

\to

(1)
Length of the tangents from A, B are 6, 7 respectively

\Rightarrow

\sqrt{x_{1}^{2} + y_{1}^{2} - a^{2}}

= 6,

\sqrt{x_{2}^{2} + y_{2}^{2} - a^{2}}

= 7

\Rightarrow

x_{1}^{2} + y_{1}^{2} - a^{2}

= 36

x_{2}^{2} + y_{2}^{2} - a^{2}

= 49

\Rightarrow

x_{1}^{2} + y_{1}^{2} + x_{1}^{2} + y_{2}^{2} - {2 a}^{2}

= 85

\Rightarrow

x_{1}^{2} + y_{1}^{2} + x_{1}^{2} + y_{2}^{2} - 2

_{1}

_{2}

+ y

_{1}

_{2}

) = 85 [From (1)]

\Rightarrow

(x_{1}^{2} + x_{2}^{2} - 2 x_{1} x_{2})

(y_{1}^{2} + y_{2}^{2} - 2 y_{1} y_{2})

= 85

\Rightarrow

(x_{1} - x_{2})^{2}

(y_{1} - y_{2})^{2}

= 85

\Rightarrow

A B^{2}

= 85

\Rightarrow

AB =

\sqrt{85}

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