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Tangents to Circles

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Understanding tangents to a circle involves recognizing their unique properties, such as touching the circle at only one point and being perpendicular to the radius at the point of tangency. This text explains how to calculate the gradient of the radius and the tangent line, and how to formulate the tangent's equation using the point-slope formula. A worked example demonstrates the application of these concepts to find the equation of a tangent line at a specific point on a circle.

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.
Close-up view of a metallic compass on white paper with a sharp point anchored at a circle's center and a pencil drawing a tangent line.

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.

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00

Unlike a secant, which intersects a circle at two points, a tangent is ______ to the radius at the point of contact.

perpendicular

01

Perpendicularity of radius and tangent

Radius to tangency point is perpendicular to tangent line at that point.

02

Gradient of radius for circle at origin

For circle centered at (0,0), gradient of radius is change in y over change in x.

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