How to Find the Tangent Line to a Circle: Step-by-Step Guide with Equations and Examples

tangent line to a circle

To find the tangent line to a circle, we need to consider both the center of the circle and the point where the tangent line intersects the circle

To find the tangent line to a circle, we need to consider both the center of the circle and the point where the tangent line intersects the circle.

Let’s assume we have a circle centered at point C(x₁, y₁) with a radius r. Let P(x, y) be a point on the circle.

To find the equation of the tangent line, we need to find the slope of the line and a point on that line.

1. Finding the slope of the tangent line:
The tangent line is perpendicular to the radius of the circle drawn from the center to the point of tangency. Since we know the slope of the radius, we can find the slope of the tangent line.

The slope of the radius (m_r) is given by:
m_r = (y – y₁)/(x – x₁)

The slope of the tangent line (m_t) is the negative reciprocal of the slope of the radius:
m_t = -1/m_r

2. Finding a point on the tangent line:
Now we need to find the coordinates of the point where the tangent line intersects the circle. To do this, we substitute the x and y coordinates of this point into the equation of the circle.

The equation of the circle (with center (x₁, y₁) and radius r) is:
(x – x₁)² + (y – y₁)² = r²

Substitute x and y with the coordinates of the point we are looking for, and solve the equation for x or y.

3. Writing the equation of the tangent line:
Using the point-slope form of a line, we can write the equation of the tangent line as:
y – y₁ = m_t (x – x₁)

Now we have the equation of the tangent line to the circle, with the slope (m_t) and a point (x₁, y₁) on the line.

Remember to substitute the values of x₁, y₁, and m_t that you have calculated from the steps above to get the specific equation of the tangent line for a given circle.

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