$T a n g e n t s a r e f r o m t h e p o i n t (h, k) t o t h e c i r c l e x^{2} + y^{2} = a^{2}; t h e n t h e t h e a r e a o f t h e t r i a n g l e f o r m e d b y t h e s t r a i g h t l i n e j o i n i n g t h e i r p o i n t s o f c o n t a c t i s \frac{a {(h^{2} + k^{2} - a^{2})}^{\frac{3}{2}}}{h^{2} + k^{2}} .$

True

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False

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Solution

The correct option is A True

Let tangents are drawn from P(h,k) to circle $x^{2} + y^{2} = a^{2}$ at point of contact of tangents be Q and R

Equation of chord of contact QR is

$h x + k y = a^{2}$

$h x + k y - a^{2} = 0 - (1)$

Draw PS perpendicular to QR

PS= $\frac{h^{2} + k^{2} - a^{2}}{\sqrt{h^{2} + k^{2}}}$

length of tangent =PR= $\sqrt{S_{1}}$

⇒PR= $\sqrt{h^{2} + k^{2} - a^{2}}$

In △PSR , $P S^{2} + S R^{2} = P R^{2}$

⇒ $S R^{2} = (\sqrt{h^{2} + k^{2} - a^{2}})^{2} - (\frac{h^{2} + k^{2} - a^{2}}{\sqrt{h^{2} + k^{2}}})^{2}$

$⟹ S R^{2} = (h^{2} + k^{2} - a^{2}) (1 - \frac{h^{2} + k^{2} - a^{2}}{h^{2} + k^{2}}$

$⟹ S R^{2} = (h^{2} + k^{2} - a^{2}) (\frac{h^{2} + k^{2} - h^{2} - k^{2} + a^{2}}{h^{2} + k^{2}}$

$⟹ S R^{2} = (h^{2} + k^{2} - a^{2}) (\frac{a^{2}}{h^{2} + k^{2}}$

$⟹ S R^{2} = \frac{a \sqrt{(h^{2} + k^{2} - a^{2})}}{\sqrt{h^{2} + k^{2}}}$

But QR=2SR

$⟹ Q R = \frac{2 a \sqrt{h^{2} + k^{2} - a^{2}}}{\sqrt{h^{2} + k^{2}}}$

Area of △PQR= $\frac{1}{2} (Q R) (P S)$

= $\frac{1}{2} \times \frac{2 a \sqrt{h^{2} + k^{2} - a^{2}}}{\sqrt{h^{2} + k^{2}}} \times \frac{h^{2} + k^{2} - a^{2}}{\sqrt{h^{2} + k^{2}}}$

Area= $\frac{a (h^{2} + k^{2} - a^{2})^{\frac{3}{2}}}{h^{2} + k^{2}}$

Suggest Corrections

Similar questions

(h, k)

x^{2} + y^{2} = a^{2}

\frac{a (h^{2} + k^{2} - a^{2})^{\frac{3}{2}}}{h^{2} + k^{2}}

\frac{x^{2}}{a^{2} + k_{1}} + \frac{y^{2}}{b^{2} + k_{1}} = 1, \frac{x^{2}}{a^{2} + k_{2}} + \frac{y^{2}}{b^{2} + k_{2}} = 1

k_{1} \neq k_{2}

h^{2} + k^{2} > a^{2}

h x + k y = a^{2}

x^{2} + y^{2} = a^{2}

\frac{x^{2}}{a^{2} + k_{1}} + \frac{y^{2}}{b^{2} + k_{1}} = 1

\frac{x^{2}}{a^{2} + k_{2}} + \frac{y^{2}}{b^{2} + k_{2}} = 1

x^{2} + y^{2} = a^{2}

h^{2} + k^{2} - a^{2} < 0

h x + k y = a^{2}

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