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.
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A point where a tangent line touches a circle at exactly one point
Radius and Tangent
The tangent line is always perpendicular to the radius of the circle at the point of tangency
A tangent only intersects a circle at one point, while a secant intersects at two points
The gradient of the radius is calculated using the coordinates of the circle's center and the point of tangency
The gradient of the tangent is the negative reciprocal of the gradient of the radius
The equation of the tangent line can be found using the point-slope form with the gradient of the tangent and the point of tangency
To find the equation of a tangent line at a specific point on a circle, the gradient of the radius and tangent must be calculated and substituted into the point-slope formula
The equation of a tangent line can be found by using the point-slope formula with the given gradient of the tangent and a point on the circle
A systematic approach of calculating the gradient of the radius, tangent, and using the point-slope formula is crucial for solving problems involving tangents to circles
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Unlike a secant, which intersects a circle at two points, a tangent is ______ to the radius at the point of contact.
perpendicular
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Perpendicularity of radius and tangent
Radius to tangency point is perpendicular to tangent line at that point.
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Gradient of radius for circle at origin
For circle centered at (0,0), gradient of radius is change in y over change in x.
