Find the equation of the tangent line to a circle at a given point.

The problem is to find the equation of the tangent line to the circle at the point .

First, we need to verify that the point lies on the circle . Substitute the coordinates of point A into the equation of the circle:

Since , the point lies on the circle .

The center of the circle is and the radius is . For circle , the center is and the radius is .

The tangent line to a circle at a point is perpendicular to the radius at that point. We need to find the slope of the radius , where is the center of the circle and is the point on the circle.
The slope of the radius is given by:

The tangent line at point A is perpendicular to the radius . The slope of the tangent line is the negative reciprocal of the slope of the radius :

Now we have the slope of the tangent line and a point on the tangent line. We can use the point-slope form of the equation of a line: , where and .

To eliminate the fraction, multiply both sides by 3:

Rearrange the equation to the standard form :

The equation of the tangent line is .

Final Answer: The final answer is