Tangents to Circles

Understanding the Concept of a Tangent to a Circle

A tangent to a circle is a line that touches the circle at precisely one point and does not intersect the circle at any other point. This single point of contact is known as the point of tangency. The tangent line is always perpendicular to the radius of the circle that passes through the point of tangency. This is a key property that distinguishes a tangent from a secant, which cuts through the circle at two points. For example, the line defined by the equation x=3 is a tangent to the circle described by x^2+y^2=9, touching it at the point (3, 0). Understanding the relationship between the tangent and the circle's radius is crucial for solving problems involving tangents.

Determining the Gradient of the Circle's Radius

To find the equation of a tangent line at a specific point on a circle, one must first determine the gradient (slope) of the radius that extends from the circle's center to the point of tangency. This radius is perpendicular to the tangent line at the point of tangency. The gradient of the radius is calculated using the coordinates of the circle's center and the point of tangency, applying the formula for the slope, which is the change in y (rise) over the change in x (run). For a circle centered at the origin (0,0) with the equation x^2+y^2=9, and a tangent point at (3, 0), the gradient of the radius is 0, since the change in y is zero.

Finding the Gradient of the Tangent Line

After determining the gradient of the radius, the next step is to find the gradient of the tangent line. The gradient of the tangent is the negative reciprocal of the gradient of the radius, due to the perpendicular relationship between the radius and the tangent at the point of tangency. If the gradient of the radius is 'm', then the gradient of the tangent will be '-1/m'. For instance, if the gradient of the radius is 1/4, the gradient of the tangent would be -4, which is the negative reciprocal of 1/4.

Formulating the Equation of the Tangent Line

With the known gradient of the tangent and the coordinates of the point of tangency, the equation of the tangent line can be formulated using the point-slope form of a linear equation, which is 'y - y1 = m(x - x1)'. Here, 'm' represents the gradient of the tangent, and '(x1, y1)' are the coordinates of the point of tangency. For a tangent that touches the circle at the point (5, 6) with a radius gradient of -1/5, the gradient of the tangent would be 5. Substituting these values into the point-slope formula yields the equation of the tangent line.

Applying the Method to a Worked Example

Consider a circle with the equation x^2+y^2=25 and a tangent that intersects the circle at the point (4, -3). The gradient of the radius, using the center (0,0) and the point of tangency (4, -3), is -3/4. The gradient of the tangent, being the negative reciprocal, is 4/3. Applying the point-slope formula 'y - y1 = m(x - x1)' with the gradient of the tangent and the point of tangency, the equation of the tangent line is derived as y + 3 = 4/3(x - 4).

Key Takeaways on Tangents of a Circle

In summary, a tangent to a circle is a line that intersects the circle at exactly one point, the point of tangency, and is perpendicular to the radius at that point. To find the equation of a tangent line, one must calculate the gradient of the radius from the circle's center to the point of tangency and then determine the gradient of the tangent as the negative reciprocal of the radius's gradient. The equation of the tangent is then established by using the point-slope formula with the tangent's gradient and the point of tangency. This systematic approach is essential for solving geometric problems involving tangents to circles.