Equations of Circles Touching Specific Lines

Understanding how to find the equation of a circle that touches specific lines is essential for solving various geometric problems. This article discusses how to determine the equation of a circle that touches the lines x 0, y 0, and x 4. We'll explore the principles behind line tangency and the relationships between the circle's center, radius, and the given lines.

Principles of Circle Tangency

A circle is tangent to a line if it touches the line at exactly one point. For a circle to touch the x-axis and the y-axis, the radius of the circle must be equal to the distances from the circle's center to these axes. Additionally, for the circle to touch the vertical line x 4, the distance between the center and this line must also be equal to the radius.

Deriving the Circle Equation

Let's consider a circle with center (h, k) and radius r. For the circle to touch the x-axis, the distance from the center to the x-axis must be r. Therefore, r k. Similarly, for the circle to touch the y-axis, the distance from the center to the y-axis must also be r. Thus, r h. Additionally, the distance from the center to the line x 4 must be equal to the radius r. Therefore, 4 - h r.

Setting Up and Solving the Equations

From the above, we have three key equations:

By equating the first two expressions for r, we get:

Substituting h k into the third equation, we get:

Solving for h gives us:

Since h k, we also have:

Thus, the center of the circle is at (2, 2) and the radius is 2.

Equation of the Circle

Using the standard form for the equation of a circle, we have:

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

Substituting the values h 2, k 2, and r 2, we get:

(x - 2)2 (y - 2)2 4

This is the equation of the circle that touches the lines x 0, y 0, and x 4.

Conclusion

There are two such circles. The second circle's center can be either (2, 2) or (2, -2). The equation of the circle with center (2, 2) is:

(x - 2)2 (y - 2)2 4

The equation of the circle with center (2, -2) is:

(x - 2)2 (y 2)2 4

Both equations describe circles with a diameter of 4 units and a radius of 2 units.