The line `y = sqrt3 x +4` touches a circle with centre at the origin. Find the radius of the circle.

Answer

To find the radius of the circle that is touched by the line \( y = \sqrt{3}x + 4 \) with its center at the origin (0, 0), we can follow these steps:

### Step 1: Identify the equation of the circle
The standard equation of a circle with center at the origin is given by:
\[ x^2 + y^2 = r^2 \]
where \( r \) is the radius of the circle.

### Step 2: Identify the slope of the line
From the given line equation \( y = \sqrt{3}x + 4 \), we can identify the slope \( m \):
\[ m = \sqrt{3} \]

### Step 3: Use the formula for the distance from the center to the line
The distance \( d \) from the center of the circle (0, 0) to the line \( Ax + By + C = 0 \) can be calculated using the formula:
\[ d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}} \]
We need to rewrite the line equation in the standard form \( Ax + By + C = 0 \):
\[ -\sqrt{3}x + y - 4 = 0 \]
Here, \( A = -\sqrt{3}, B = 1, C = -4 \).

### Step 4: Substitute into the distance formula
Substituting \( x_1 = 0 \) and \( y_1 = 0 \) into the distance formula:
\[ d = \frac{|-4|}{\sqrt{(-\sqrt{3})^2 + 1^2}} = \frac{4}{\sqrt{3 + 1}} = \frac{4}{\sqrt{4}} = \frac{4}{2} = 2 \]

### Step 5: Relate the distance to the radius
Since the line touches the circle, the distance \( d \) from the center of the circle to the line is equal to the radius \( r \):
\[ r = d = 2 \]

### Conclusion
Thus, the radius of the circle is:
\[ \boxed{2} \]