A tangent is a line that touches a circle at only one point on its circumference. Unlike a secant, which intersects with two points on the circle, a tangent aligns with the circle at that single point.

If you need to find the equation of a tangent to a circle, the first step is to determine the gradient of the radius. This can be done by using the coordinates of the circle's center and the point where the tangent intersects the circle. The gradient of the radius acts as a normal line, perpendicular to the tangent.

To find the gradient of the tangent, simply take the negative reciprocal of the gradient of the radius. This will give you the gradient of the tangent line to the circle.

For example, let's say we have a circle with the equation x² + y² = 25 and a tangent that intersects the circle at the point (3, 0). By substituting the coordinates of the center (0, 0) and the exterior point (3, 0) into the gradient formula, we can determine that the gradient of the radius is 1/3. The negative reciprocal of this, -3, is then the gradient of the tangent line.

Once you have the gradient of the tangent, you can use one of three possible equations to find the equation of the tangent. The first two formulas are typically easier to use, but if you are asked to put your final answer into the third form, you can use the first or second formula and then rearrange the equation accordingly.

For instance, if we have a tangent line that intersects circle A at point (5, 6) and the gradient of the radius is 5, we can use the first or second formula to find the equation of the tangent and then rearrange it into the third form if necessary.

In a visual representation, the process of finding the equation of a tangent to a circle can be seen in the diagram below:

**Key Takeaways from Finding the Equation of a Tangent to a Circle:**

- A tangent to a circle is a line that touches the circumference at only one point.
- The equation of a tangent can be found by using the coordinates of the center and the point of intersection.
- The gradient of the radius acts as a normal line, perpendicular to the tangent.
- The negative reciprocal of the gradient of the radius gives us the gradient of the tangent line.
- There are three different formulas for finding the equation of a tangent, with the first two being typically easier to use.

In conclusion, determining the equation of a tangent to a circle involves finding the gradient of the radius and using it to determine the gradient of the tangent. This gradient can then be substituted into one of three possible formulas to give the equation of the tangent line to the circle.

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