A tangent to a circle is a straight line that touches the circle at just one point. To find the equation of the tangent to a circle from an external point, we need to follow several steps. Below is a detailed explanation:

1. Understanding the Circle’s Equation

The general equation of a circle with center (h, k) and radius r is given by:

(x - h)² + (y - k)² = r²

2. Identifying the External Point

Let’s define the external point from which the tangent will be drawn as P(x₁, y₁). This point should be outside the circle, meaning its distance from the circle’s center (h, k) must be greater than the circle’s radius (r).

3. Finding the Slope of Tangent Lines

To find the tangent line, we first need the slope of the line connecting point P to the center of the circle. The slope m can be calculated as:

m = (y₁ - k) / (x₁ - h)

4. Using the Point-Slope Form of the Line

Now, we can use the point-slope form of a line to write the equations of the tangent lines:

y - y₁ = m(x - x₁)

5. Finding the Tangent Points

To determine the specific points of tangency (let’s call them T(X₀, Y₀) on the circle), we need to solve the system of equations formed by:

(x - h)² + (y - k)² = r²

y - y₁ = m(x - x₁)

6. Solving the Equations

By substituting the linear equation of the tangent into the circle’s equation, we can find the coordinates of points T(X₀, Y₀). This will require rearranging and potentially using the quadratic formula to find the solutions. There will typically be two solutions, corresponding to the two tangents (if they exist).

7. Final Equation of the Tangent(s)

Once we have the point(s) of tangency, we can then substitute back into the point-slope equation obtained earlier:

y - Y₀ = m(x - X₀)

This gives us the equation of the tangent at that particular point (X₀, Y₀).

Example

Consider a circle centered at (2, 3) with a radius of 3 and an external point P(5, 5). The equation of the circle is:

(x - 2)² + (y - 3)² = 9

1. Find the slope m:

m = (5 - 3) / (5 - 2) = 2/3

2. Write the tangent line equation:

y - 5 = (2/3)(x - 5)

3. Substitute into the circle’s equation and solve for the points of tangency.

In conclusion, the equation of the tangents can be derived systematically using the methods outlined above. It’s important to visually understand the geometry involved to enhance comprehension.