If P is a point on a circle with center C, then the tangent...

If $P$ is a point on a circle with center $C$, then the tangent line to the circle at $P$ is the straight line through $P$ that is perpendicular to the radius $C P$. In Exercises $67-70$, find the equation of the tangent line to the circle at the given point. $x^{2}+y^{2}=169$ at (-5,12)

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Key Concepts

Circle Equation

A circle equation in standard form, like x² + y² = r², represents all points at a constant distance (radius) from a fixed center point. Understanding this equation involves recognizing that the center and the radius are key geometric characteristics that define the circle's overall shape and position on the coordinate plane.

Radius

The radius of a circle is a line segment connecting the center of the circle to any point on its circumference. It plays a central role in many geometric properties, such as defining the circle's size and being orthogonal (perpendicular) to the tangent line at the point of contact.

Tangent Line

A tangent line to a circle is a line that touches the circle at exactly one point, called the point of tangency. The key property of a tangent line is that it is perpendicular to the radius drawn to the point of contact, which is fundamental in determining its slope and equation.

Perpendicular Lines

In coordinate geometry, perpendicular lines have slopes that are negative reciprocals of each other. This property is critical when finding the equation of a tangent line to a circle, as the tangent's slope is computed by taking the negative reciprocal of the slope of the radius connecting the circle's center with the point of tangency.

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