Deriving the Equation for the Tangent at a Point from an External Point to a Circle

In this article, we will explore the process of deriving the equation for finding the tangent from an external point to a circle. This will involve several step-by-step procedures to ensure a clear and systematic approach.

Introduction

Understanding how to derive the equation for the tangent at a point from an external point to a circle is fundamental in both theoretical and practical applications of geometry. This process involves algebraic manipulation and geometric principles.

Step-by-Step Guide

Step 1: Define the Circle and External Point

Consider a circle with the center at the point (h, k) and radius r. The equation of the circle can be expressed as:

(x - h)2 (y - k)2 r2

Suppose we have an external point P with coordinates (x?, y?) from which we want to draw tangents to the circle.

Step 2: Condition for Tangency

A line is tangent to the circle if the distance from the center (h, k) to the line equals the radius r. This ensures that the line touches the circle at exactly one point.

Step 3: Equation of the Line

The general equation of a line passing through point P can be written in slope-intercept form as:

y - y? m(x - x?)

Here, m is the slope of the line.

Step 4: Substitute the Line Equation into the Circle Equation

To find the points of intersection between the line and the circle, substitute the line equation into the circle equation:

Rearranging the line equation gives:

y mx - mx? y?

Substitute this into the circle equation:

(x - h)2 ((mx - mx? y?) - k)2 r2

This will result in a quadratic equation in terms of x:

Ax2 Bx C 0

Where A, B, and C are coefficients that depend on m, h, k, r, x?, and y?.

Step 5: Expand and Rearrange

Expand the equation and rearrange it to form a quadratic equation:

Ax2 Bx (C) 0

Step 6: Set the Discriminant to Zero

To find the distance condition for tangency, set the discriminant of the quadratic equation to zero:

D B2 - 4AC 0

This condition ensures that there is exactly one solution for x, meaning the line touches the circle at exactly one point.

Step 7: Solve for the Slope m

Solving the equation D 0 will give you the possible values for the slope m. You can express it in terms of the coordinates of point P and the center of the circle.

Step 8: Find the Tangent Points

Once you have the slopes, use them to find the equations of the tangent lines by substituting m back into the line equation:

y - y? mx - x?

This results in one or two equations for the tangent lines from point P to the circle.

Summary

The process of deriving the equation for the tangent at a point from an external point to a circle involves defining the circle and external point, finding the general equation of the line, substituting the line equation into the circle equation, and setting the discriminant to zero. Finally, solve for the slope and use it to write the equations of the tangent lines.

Write the circle equation.

Write the lines equation through the external point.

Substitute the line equation into the circle equation.

Set the discriminant of the resulting quadratic to zero.

Solve for the slopes and write the tangent line equations.

This method provides a systematic approach to finding the tangents from an external point to a circle, making it an essential skill in advanced geometry and trigonometry.