First slide

Definition of a circle

Question

The area of the triangle, in square units, formed by the tangents from the point $(4, 3)$ to the circle $x^{2} + y^{2} = 9$ and the line joining their points of contact, is

Moderate

A

$\frac{25}{192}$
B

$\frac{192}{25}$
C

$\frac{384}{25}$
D

none of these

Solution

We know that the area $∆$ of the triangle formed by the tangent drawn from $(x_{1}, y_{1})$ to the circle $x^{2} + y^{2} = a^{2}$ and their chord of contact is given by
$Δ = \frac{a {(x_{1}^{2} + y_{1}^{2} - a^{2})}^{3 / 2}}{x_{1}^{2} + y_{1}^{2}}$
Here, the point is $P (4, 3)$ and the circle is $x^{2} + y^{2} = 9 .$
$∴$ Required area $= \frac{3 {(4^{2} + 3^{2} - 9)}^{3 / 2}}{4^{2} + 3^{2}}$ sq. units $= \frac{192}{25}$ sq. units.

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