Let the tangents at two point A and B on the circle x2+y2-4x +3=0 meet at origin O (0,0). Then the area of the triangle of OAB is

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Let the tangents at two point A and B on the circle $x^{2} + y^{2} - 4 x + 3 = 0$ meet at origin O (0,0). Then the area of the triangle of OAB is

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$\frac{3}{2 \sqrt{3}}$

$\frac{3}{4 \sqrt{3}}$

$\frac{3 \sqrt{3}}{4}$

$\frac{3 \sqrt{3}}{2}$

answer is B.

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Detailed Solution

The given circle is $x^{2} + y^{2} - 4 x + 3 = 0$

Equation of chord of contact of origin is $S_{1} = 0$

Hence the equation of the chord of contact is $2 x = 3$

The length of the tangent from origin to the circle is $\sqrt{S_{11}}$

Hence, $O A = O B = \sqrt{3}$

Distance from origin to the chord of contact is $O M = \frac{3}{2}$

Area of the triangle $O A B$ is $\frac{1}{2} (2 A M) (O M)$

Therefore, the area of the triangle $O A B$ is $\frac{1}{2} \cdot \sqrt{3} \cdot \frac{3}{2} = \frac{3 \sqrt{3}}{4}$

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