The tangent and the normal lines at the point ( 3 , 1) to the...

The tangent and the normal lines at the point

\left(\sqrt{3}, 1\right)

to the circle

{x}^{2} + {y}^{2} = 4

and the

x

-axis form a triangle. The area of this triangle (in square units) is: The tangent and the normal lines at the point ( 3 , 1) to the circle x ^ 2 + y ^ 2 = 4 and the x -axis form a triangle.

The tangent and the normal lines at the point ( 3 , 1) to the circle x ^ 2 + y ^ 2 = 4 and the x -axis form a triangle.

Options

A $\dfrac{1}{3}$
B $\dfrac{2}{\sqrt{3}}$
C $\dfrac{4}{\sqrt{3}}$
D $\dfrac{1}{\sqrt{3}}$

Correct answer

B. $\dfrac{2}{\sqrt{3}}$

Step-by-step solution

Equation tangent at

P\left(\sqrt{3}, 1\right)

x ⋅ \sqrt{3} + y ⋅ 1 = 4

x-intercept of this tangent is

\dfrac{4}{\sqrt{3}}

.
Hence, area of

\Delta = \dfrac{1}{2} ⋅ \dfrac{4}{\sqrt{3}} ⋅ 1 = \dfrac{2}{\sqrt{3}}

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