Finding the Radius and Standard Equation of a Circle Tangent to a Line

In this article, we will walk through the process of finding the radius and the standard equation of a circle that is tangent to a given line. We explore the steps to measure the distance from the circle's center to the line and then deduce the circle's equation. This is a practical application of geometric principles and involves some algebraic calculations. Let's begin!

Step 1: Measure the Distance from the Circle's Center to the Line

Given a line 3x - 4y 4 0 and a circle with its center at (1, -7), we need to find the distance from the center of the circle to the line. This distance will be the radius of the circle since the circle is tangent to the line.

Distance Formula

To find the distance from a point to a line, we use the formula:

D |Ax0 By0 C| / √A^2 B^2

Where:

A 3 B -4 C 4 x0 1 y0 -7

Calculations

Substituting the given values into the formula:

D |3times;1 - 4times;(-7) - 4| / √3^2 (-4)^2

D |3 28 - 4| / √9 16

D |27| / √25

D 27 / 5 5.4

The distance from the center of the circle to the line is 5.4 units.

Step 2: Determine the Radius

Since the circle is tangent to the line, the distance we calculated is the radius of the circle.

Radius (r) 5.4

Step 3: Write the Standard Equation of the Circle

The standard equation of a circle with the center at (h, k) and a radius of r is:

(x - h)^2 (y - k)^2 r^2

Substituting the center of the circle at (1, -7) and the radius 5.4 into the standard equation:

(x - 1)^2 (y 7)^2 5.4^2

(x - 1)^2 (y 7)^2 29.16

Summary

Radius: 5.4 units

Standard Equation: (x - 1)^2 (y 7)^2 29.16

The standard equation of the circle is a powerful formula that allows us to represent geometric shapes algebraically. Understanding how to apply the distance formula to find the radius and then using the standard equation is crucial for any circle-related problem.

Key Takeaways:

Understanding the distance formula is essential for solving many geometric problems. The standard equation of a circle is invaluable for representing circles in an algebraic form. A practical exercise in geometry involving tangent lines and circles can greatly enhance your problem-solving skills.

Conclusion

We have explored the detailed steps to find the radius and the standard equation of a circle that is tangent to a given line. By using the distance formula and the standard equation of a circle, we can solve such problems efficiently and accurately. Practice these steps and apply them to different scenarios to build a strong foundation in geometrical problem-solving.