Introduction to the Problem

Understanding how to find the equation of a circle that is tangent to a given line at a specific point is a fundamental concept in geometry and algebra. This article will walk you through the process using a specific example: a line defined by y x is tangent to a circle at the point (2, 2).By the end of this article, you will have a clear understanding of the steps involved and be able to solve similar problems on your own.

Steps to Solve the Problem

We will follow a series of meticulous steps to determine the equation of the circle. Each step is designed to build upon the previous one, making the solution intuitive and easy to follow. Let's begin with the fundamentals and work our way through the problem.

Step 1: Identify the Center of the Circle

Given that the line y x is tangent to the circle at the point (2, 2), the radius of the circle at this point is perpendicular to the tangent line. The slope of the tangent line y x is 1. Therefore, the slope of the radius, which is perpendicular to the tangent line, is -1. This helps us find the center of the circle.

Step 2: Find the Center of the Circle

The center of the circle can be expressed as (h, k). Since the radius is perpendicular to the tangent line, the center must lie along the line with a slope of -1 that passes through the point (2, 2). The equation of this line can be written as:

y - 2 -1(x - 2) rarr; y -x 4

Step 3: Determine the General Form of the Circle's Equation

The general equation of a circle is given by:

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

where (h, k) is the center and r is the radius.

Step 4: Use the Point of Tangency

Since the point (2, 2) lies on the circle, we can substitute x 2 and y 2 into the circle's equation:

(2 - h)2 (2 - k)2 r2

Step 5: Find the Radius

The distance from the center (h, k) to the point (2, 2) is the radius r. Additionally, the distance from the center (h, k) to the line y x must also equal r. The distance d from a point (h, k) to the line ax by c 0 is given by:

d |ah bk c| / √(a2 b2)

For the line y x, which can be rewritten as x - y 0 or 1x - 1y 0 0, we have:

-a 1, -b -1, -c 0

Thus, the distance from (h, k) to the line is:

d |h - k| / √(12 (-1)2) |h - k| / √2

Setting the distances equal, we get:

r |h - 2|2 |k - 2|2]

and

r |h - k| / √2

Step 6: Equating the Two Expressions for r

Substituting k from the line equation (k -h 4), we find:

r |h - 2|2 |-h 4 - 2|2] |h - 2|2 |-h 2|2]

Simplifying, we get:

r |h - 2|2 |h - 2|2]] √(2|h - 2|) h - 2√2

Equating the two expressions for r:

(h - (-h 4)) / √2 h - 2√2

This simplifies to:

(2h - 4) / √2 h - 2√2

Step 7: Solving for h

From here, we can solve for h and subsequently find k. After solving, we find:

h 3, k 1

Step 8: Finding the Equation of the Circle

Substituting (h, k) back into the circle's equation:

(x - 3)2 (y - 1)2 r2

Since r √2, we get:

(x - 3)2 (y - 1)2 2

Conclusion

The equation of the circle that is tangent to the line y x at the point (2, 2) is:

(x - 3)2 (y - 1)2 2

This article has provided a step-by-step guide to solving this problem. Understanding these steps will help you tackle similar geometric and algebraic problems with ease.