Question

The area of the triangle 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

• A. $\frac{25}{192}sq.units$
• B. $\frac{192}{25}sq.units$
• C. $\frac{384}{25}sq.units$
• D. None

Answer 1

Answer: (B)

$\frac{192}{25}sq.units$

Answer 2

The equation of the chord of contact of tangents drawn from P (4, 3) to $x^{2} + y^{2} = 9$ is $4x + 3y = 9.$ The equation of OP is $y = \frac{3}{4}x.$ Now, OM = (length of the perpendicular from (0, 0) on $4x + 3y - 9 = 0) = \frac{9}{5}$

$\therefore$ $QR = 2.QM = 2\sqrt{OQ^{2} - OM^{2}} = 2\sqrt{9 - \frac{81}{25}} = \frac{24}{5}$

Now, $PM = OP - OM = 5 - \frac{9}{5} = \frac{16}{5}$.

So, Area of $\Delta PQR = \frac{1}{2}\left( \frac{24}{5} \right)\left( \frac{16}{5} \right) = \frac{192}{25}sq.units$