Question

\(\approx\) 50. A belt fits tightly around the two circles, with equations \((x-1)^{2}+(y+2)^{2}=16\) and \((x+9)^{2}+(y-10)^{2}=16\) How long is this belt?

Short answer

The belt length is approximately 26.84 units.

Step-by-step solution

Understand the Problem

We are given two circles with equations \((x-1)^2 + (y+2)^2 = 16\) and \((x+9)^2 + (y-10)^2 = 16\). Both circles have the same radius of 4 units (since the radius\(r\) is the square root of 16). The centers of the circles are \((1, -2)\) and \((-9, 10)\). We need to find the length of the belt that fits tightly around both circles, which is the external tangential distance between the circles.

Calculate Center Separating Distance

To find the distance between the centers of the circles, use the distance formula: \[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Substitute \(x_1 = 1, y_1 = -2, x_2 = -9, y_2 = 10\):\[D = \sqrt{(-9-1)^2 + (10-(-2))^2} = \sqrt{(-10)^2 + (12)^2} = \sqrt{100 + 144} = \sqrt{244} = 2\sqrt{61}\]

Calculate External Tangent Length

The formula for the length of the belt (external tangents) is:\[L = \sqrt{d^2 - (r_1 + r_2)^2}\] where \(d\) is the distance between the centers and \(r_1 = r_2 = 4\) are the radii.Given \(d = 2\sqrt{61}\):\[L = \sqrt{(2\sqrt{61})^2 - (4 + 4)^2}\]\[L = \sqrt{244 - 64} = \sqrt{180} = \sqrt{36 \times 5} = 6\sqrt{5} \approx 13.42\]

Conclusion: Total Belt Length

The complete belt length includes both external tangents. Since we have two identical tangent segments, the total belt length is twice the single external tangent length:\[2 \times 6\sqrt{5} = 12\sqrt{5} \approx 26.84\]

Key concepts

Analytic Geometry

Analytic geometry provides a bridge between algebra and geometry, allowing us to solve geometrical problems using algebraic equations. It deals with points, lines, and curves in a coordinate plane.
This exercise gives us two circles described by the equations \((x-1)^2 + (y+2)^2 = 16\) and \((x+9)^2 + (y-10)^2 = 16\). Each equation represents a circle in this coordinate plane. The first equation centers the circle at \((1, -2)\), while the second centers it at \((9, -10)\).
The radius can be extracted from the right-hand side of the equations, where \(16\) represents the square of the radius, giving these circles a radius of 4 units each. Understanding the placement and size of these shapes is crucial for determining tangents and lengths between them.

Distance Formula

The distance formula is a fundamental concept in both geometry and algebra. It calculates the distance between two points in a plane. This formula is derived from the Pythagorean theorem and is expressed as \[D = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\].
In our problem, it helps determine the distance between the centers of the two circles, \((1, -2)\) and \((-9, 10)\). By substituting the coordinates into the formula, we find that \[D = \sqrt{(-10)^2 + (12)^2} = \sqrt{244} = 2\sqrt{61}\].
This value is crucial as it later aids in calculating the length of the external tangents. Understanding the distance between the circle centers allows us to progress towards solving the given problem.

External Tangents

External tangents are lines that touch both of two circles without crossing the space between them. Each circle in this problem is tangent to the belt that wraps around both.
For two equal circles, like those in this exercise, the formula for the external tangent is \[L = \sqrt{d^2 - (r_1 + r_2)^2}\].
Here, \(d\) is the center-to-center distance and \(r_1\) and \(r_2\) are the radii of the circles.
Substituting \(d = 2\sqrt{61}\) and \(r_1 = r_2 = 4\), the calculation becomes \[L = \sqrt{244 - 64} = \sqrt{180} = 6\sqrt{5} \approx 13.42\].
The length of the external tangent is essential for computing the total length of the belt that fits around both circles.

Problem Solving Steps

Problem-solving in mathematics is a structured approach to finding solutions efficiently. This method often starts with understanding the problem, applying relevant mathematical concepts, and logically arriving at the solution.
In this exercise:

• Step 1: We first determined the centers and radii of the circles from their equations.
• Step 2: Using the distance formula, we calculated the separation between these centers.
• Step 3: We used the calculated distance to find the length of a single external tangent using the tangent formula.
• Step 4: Lastly, we recognized that the complete belt would consist of two external tangents, so we doubled the length of one to find \(12\sqrt{5} \approx 26.84\).

Each step builds on the last, ensuring the problem is handled methodically and accurately.