If a circle of radius 3 units is touching the lines `sqrt3 y^2 - 4xy +sqrt3 x^2 = 0` in the first quadrant then length of chord of contact to this circle is :

Options

A. `(sqrt3 +1)/(2)`
B. `(sqrt3+1)/(sqrt2)`
C. `3((sqrt3+1)/(sqrt2))`
D. `(3(sqrt3+1))/(2)`

Explanation

To find the length of the chord of contact from a circle of radius 3 units that touches the lines given by the equation \( \sqrt{3}y^2 - 4xy + \sqrt{3}x^2 = 0 \) in the first quadrant, we can follow these steps:

### Step 1: Factor the given equation
The equation \( \sqrt{3}y^2 - 4xy + \sqrt{3}x^2 = 0 \) can be factored. We can rewrite the middle term as follows:

\[ \sqrt{3}y^2 - 4xy + \sqrt{3}x^2 = \sqrt{3}y^2 - 3xy - xy + \sqrt{3}x^2 \]

Now, grouping the terms:

\[ = \sqrt{3}y(y - \frac{4}{\sqrt{3}}x) + x(\sqrt{3}x - 4y) = 0 \]

This can be factored as:

\[ (y - \sqrt{3}x)(\sqrt{3}y - x) = 0 \]

### Step 2: Identify the lines
From the factorization, we have two lines:

1. \( y = \sqrt{3}x \)
2. \( y = \frac{x}{\sqrt{3}} \)

### Step 3: Determine the angles of the lines
The slopes of the lines give us the angles they make with the x-axis:

- For \( y = \sqrt{3}x \), the slope is \( \sqrt{3} \), which corresponds to an angle of \( 60^\circ \).
- For \( y = \frac{x}{\sqrt{3}} \), the slope is \( \frac{1}{\sqrt{3}} \), which corresponds to an angle of \( 30^\circ \).

### Step 4: Analyze the angle between the lines
The angle between these two lines can be calculated as:

\[ \text{Angle between the lines} = 60^\circ - 30^\circ = 30^\circ \]

### Step 5: Find the angle of the radius to the chord of contact
The angle between the radius to the point of contact and the line joining the center of the circle to the origin (which is at the angle of \( 0^\circ \)) is half of the angle between the lines. Thus, it is:

\[ \theta = \frac{30^\circ}{2} = 15^\circ \]

### Step 6: Use trigonometry to find the length of the chord
In triangle \( AMC \) (where \( A \) is the center of the circle, \( M \) is the midpoint of the chord, and \( C \) is the point of contact), we have:

\[ \cos(15^\circ) = \frac{AM}{3} \]

Thus,

\[ AM = 3 \cos(15^\circ) \]

Using the cosine value:

\[ \cos(15^\circ) = \frac{\sqrt{3} + 1}{2\sqrt{2}} \]

So,

\[ AM = 3 \cdot \frac{\sqrt{3} + 1}{2\sqrt{2}} = \frac{3(\sqrt{3} + 1)}{2\sqrt{2}} \]

### Step 7: Find the length of the chord \( AB \)
The length of the chord \( AB \) is twice the length of \( AM \):

\[ AB = 2 \cdot AM = 2 \cdot \frac{3(\sqrt{3} + 1)}{2\sqrt{2}} = \frac{3(\sqrt{3} + 1)}{\sqrt{2}} \]

### Final Answer
The length of the chord of contact is:

\[ \frac{3(\sqrt{3} + 1)}{\sqrt{2}} \]