Finding the Equation of a Tangent to a Circle: Step-by-Step Guide with Example

tangent to a circle

A tangent to a circle is a line that touches the circle at exactly one point, called the point of tangency

A tangent to a circle is a line that touches the circle at exactly one point, called the point of tangency. The tangent line is perpendicular to the radius of the circle at the point of tangency.

To find the equation of a tangent to a circle at a given point, you would need the coordinates of the center of the circle and the coordinates of the point of tangency.

Here’s an example to illustrate the process:

Suppose we have a circle with center (2, -3) and a radius of 4. We want to find the equation of the tangent to the circle at the point (6, -3).

Step 1: Determine the slope of the radius at the point of tangency.
The radius of the circle is a line segment connecting the center of the circle to the point of tangency. The slope of this radius can be found using the formula:
slope = (y2 – y1) / (x2 – x1)

Using the coordinates (2, -3) and (6, -3), we substitute the values into the formula:
slope = (-3 – (-3)) / (6 – 2)
= 0 / 4
= 0

The slope of the radius at the point of tangency is 0.

Step 2: Find the slope of the tangent line.
Since the radius and the tangent line are perpendicular, their slopes are negative reciprocals of each other. Therefore, the slope of the tangent line is the negative reciprocal of 0, which is undefined.

Step 3: Write the equation of the tangent line in point-slope form.
The equation of a line in point-slope form is given by:
y – y1 = m(x – x1)
where (x1, y1) are the coordinates of the point of tangency and m is the slope of the tangent line.

Using the point (6, -3) and the slope (undefined), we substitute the values into the equation:
y – (-3) = undefined(x – 6)

This simplifies to:
y + 3 = undefined

The equation of the tangent line to the circle at the point (6, -3) is y + 3 = undefined.

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