Problem 75 - Algebra 2

Question

Find the distance between the centers of the circles with equations \((x-5)^{2}+(y-1)^{2}=16\) and \((x+1)^{2}+(y-9)^{2}=49\)

Step-by-Step Solution

Answer

The distance between the centers of the circles is 10 units.

Step 1: Identify the Centers of the Circles

The centers of the circles can be determined by looking at the coefficients in the circle equations. If \((x-a)^2 + (y-b)^2 = r^2\) is the equation of a circle, \((a, b)\) is the center of the circle. Thus, for the circle with equation \((x-5)^{2}+(y-1)^{2}=16\), the center is \((5, 1)\), and for the circle with equation \((x+1)^{2}+(y-9)^{2}=49\), the center is \((-1,9)\).

Step 2: Apply the Distance Formula

You can now calculate the distance between the two centers using the distance formula. This formula is \(\sqrt{(x_{2}-x_{1})^2 + (y_{2}-y_{1})^2}\). In this case, \((x_{1}, y_{1})\) are the coordinates of the center of the first circle, \((5, 1)\) and \((x_{2}, y_{2})\) are the coordinates of the center of the second circle, \((-1, 9)\).

Step 3: Calculate the Distance

Substituting these values into the distance formula, we have \(\sqrt{((-1)-5)^2 + (9-1)^2} = \sqrt{(-6)^2 + 8^2 = 36 + 64} = \sqrt{100}\).

Key Concepts

Center of a Circle

Understanding the center of a circle is vital when dealing with circle equations. In any standard circle equation of the form \[ (x-a)^2 + (y-b)^2 = r^2 \] the point \((a, b)\) denotes the center. Here, \(a\) and \(b\) are the values subtracted from the \(x\) and \(y\) terms, respectively. This equation implies that the circle is plotted around the center at \((a, b)\) with a radius of \(r\). Furthermore, this center serves as a reference for various circle properties like the diameter, circumference, and position in the Cartesian plane.
Using the given exercise as an example, let's identify the centers for the circles involved:

• The equation \((x-5)^2 + (y-1)^2 = 16\) reveals that the center is at \((5, 1)\).
• Similarly, for \((x+1)^2 + (y-9)^2 = 49\), the center is \((-1, 9)\).

Recognizing these centers allows us to move forward to use other formulas, such as the distance formula, to find spatial relationships between shapes in a plane.

Distance Formula

The distance formula is a handy tool to calculate the straight-line distance between two points in a coordinate plane. If given two points, \((x_1, y_1)\) and \((x_2, y_2)\), the formula is expressed as:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] This formula is derived from the Pythagorean theorem and is crucial in geometry and algebra. It allows for the measurement of how far apart two locations are, which is particularly useful in various applications, such as navigation, physics, and engineering.
Applying this formula to the centers we found:

• The center of the first circle is \((5, 1)\)
• The center of the second is \((-1, 9)\)

Plugging these into the formula:\[ d = \sqrt{((-1)-5)^2 + (9-1)^2} \]This simplifies to:\[ d = \sqrt{(-6)^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10 \]Thus, the distance between the centers of these two circles is 10 units.

Circle Equations

An equation of a circle describes all the points that make up a circle in a coordinate plane. The general form of this equation is:\[ (x-a)^2 + (y-b)^2 = r^2 \] In this format:

• \((a, b)\) represents the center of the circle.
• \(r\) is the radius, which is the distance from the center to any point on the circle.

These equations allow you to understand the size and location of a circle. The circle's circumference encloses all the points that satisfy this equation. This mathematical representation is a powerful tool in geometry to analyze circular objects and their properties.
In the example provided, each equation gives insight into the respective circle’s shape:

• The equation \((x-5)^2 + (y-1)^2 = 16\) shows a circle with a radius 4, centered at \((5, 1)\).
• Similarly, \((x+1)^2 + (y-9)^2 = 49\) indicates a circle with a radius 7, centered at \((-1, 9)\).

Understanding circle equations thus simplifies the process of visualizing and interpreting circular forms in various scientific and mathematical contexts.