3. The line $3 x-y-6=0$ is a tangent to a circle whose center is at point $C(6,2)$. Calculate the radius of this circle. The result should be exact.
3. We simplify the equation of the line to $y=3 x-6$. We write the slope $k=3$ and consider that the tangent is perpendicular to the radius, which means the slope of the radius line is $k=-\frac{1}{3}$. We consider that the radius line passes through the center of the circle $C(6,2)$ and determine the equation of the line, which is $y=-\frac{x}{3}+4$. The point of tangency between the tangent and the circle is the intersection of the lines $y=3 x-6$ and $y=-\frac{x}{3}+4$. We write the equation $3 x-6=-\frac{x}{3}+4$. We calculate $P(3,3)$. The radius is the distance between the center and the point of tangency of the tangent on the circle. We use the formula for the distance between two points $d(A, B)=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$, insert the appropriate
coordinates and calculate $r=\sqrt{10}$.

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-12.jpg?height=60&width=1644&top_left_y=313&top_left_x=203)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-12.jpg?height=68&width=1639&top_left_y=363&top_left_x=206)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-12.jpg?height=63&width=1639&top_left_y=414&top_left_x=206)

The equation for calculating the intersection $3 x-6=-\frac{x}{3}+4 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-12.jpg?height=49&width=1636&top_left_y=518&top_left_x=210)

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-12.jpg?height=60&width=1639&top_left_y=570&top_left_x=206)

Using the distance formula or inserting the data ......................................................................................

![](https://cdn.mathpix.com/cropped/2024_06_07_7bf3e9b12130e254269ag-12.jpg?height=60&width=1639&top_left_y=661&top_left_x=206)