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.) $x^{2}+y^{2}+6 x-8 y+15=0$ at (-2,1)

Key Concepts

Tangent Line to a Circle

A tangent line to a circle is a line that touches the circle at exactly one point. This property is significant because the tangent line at any point on the circle is perpendicular to the radius drawn to that point, a fact that is used to determine the tangent line’s slope and subsequently, its equation.

Perpendicularity (Slopes of Lines)

When two lines are perpendicular, the product of their slopes is -1. This concept is essential in finding the tangent line to a circle, as the tangent line is perpendicular to the radius at the point of tangency. By finding the slope of the radius, one can easily compute the slope of the tangent line by taking the negative reciprocal.

Equation of a Circle

The equation of a circle in the coordinate plane is normally given in the form (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is its radius. When a circle is presented in a more general quadratic form, it represents the circle’s equation and can be manipulated to reveal its center and radius.

Completing the Square

Completing the square is an algebraic technique used to transform a quadratic expression into a perfect square trinomial. This technique is crucial when rewriting the general equation of a circle into its standard form, which then allows for the easy identification of the circle’s center and radius.

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