How to Find the Radii of Two Circles with a Centre Point at the Origin That Touch a Given Circle

Mathematically determining the radii of circles with a center at the origin that touch a given circle can be an intriguing exercise in geometry. In this article, we explain the steps to solve such a problem and provide a detailed analysis of the process, including the application of the distance formula and the conditions of tangency.

Understanding the Problem

Given a circle with the equation:

[x^2 y^2 - 8x - 6y 24 0]

The objective is to find the radii of two circles centered at the origin (0,0) that touch the given circle. Let's walk through the steps to solve this problem.

Step 1: Rewrite the Circle Equation

To solve this problem, we first need to rewrite the given circle equation in standard form. This process involves completing the square for both x and y.

Completing the Square

For (x):

[x^2 - 8x (x - 4)^2 - 16]

For (y):

[y^2 - 6y (y - 3)^2 - 9]

Substituting these back into the equation:

[(x - 4)^2 - 16 (y - 3)^2 - 9 24 0]

Simplifying this, we get:

[(x - 4)^2 (y - 3)^2 - 1 0]

[(x - 4)^2 (y - 3)^2 1]

Step 2: Identify the Center and Radius of the Given Circle

The standard form of a circle is:

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

From the standard form, we can see:

The center of the given circle is (4, 3). The radius (r) is (1).

Step 3: Calculate the Distance from the Origin

The distance (d) from the origin (0,0) to the center (4,3) is given by the distance formula:

[d sqrt{(4 - 0)^2 (3 - 0)^2} sqrt{16 9} sqrt{25} 5]

Step 4: Determine the Radii of the Circles

Let (R) be the radius of the circles centered at the origin. For the circles to touch the given circle, the relationship between the radius (R) and the distance (d) must satisfy:

For external tangency:

[R 1 5 quad Rightarrow quad R 4]

For internal tangency:

[5 - R 1 quad Rightarrow quad R 4]

Conclusion: Both conditions yield the same radius. Therefore, the radius of the circles centered at the origin that touch the given circle is 4.

Additional Considerations

In a more complex scenario, the given circle equation can be modified to include coordinates beyond 4 and 3. For example:

Given the equation[x^2 y^2 - 8x - 6y 24 0]

1. By completing the square, we find that the circle's center is (4, 3) and radius is 1.

2. The line (y frac{3}{4}x) passes through the origin (0,0) and intersects the circle at point P(18/5, 18/5) or (12/5, 12/5).

3. For the first circle with radius 6 centered at (18/5, 18/5), its radius (R_1) is 6.

4. For the second circle with radius 4 centered at (12/5, 12/5), its radius (R_2) is 4.

This analysis shows that the process can be extended to more complex configurations and equations, allowing for a deeper understanding of the geometric relationships involved.

Key Points:

The given circle's equation is rewritten in standard form. The distance from the origin to the center of the circle is calculated. The conditions of tangency are used to determine the radii.

In conclusion, understanding the radii of circles that touch a given circle involves a combination of algebraic manipulation and geometric principles. Whether the tangency is external or internal, the method remains consistent, providing a valuable tool for solving similar problems in mathematics.