Problem 49 - Algebra 2

Question

Use the given information to write an equation of the circle. center \((2,1),\) through \((6,4)\)

Step-by-Step Solution

Answer

\((x-2)^2 + (y-1)^2 = 25\

Step 1: Identify the coordinates of the center

The coordinates for the center of the circle are provided as \( (h, k) = (2,1) \).

Step 2: Calculate the radius

A point on the circle is given as \((6,4)\). The radius of the circle can be calculated using the distance formula where \(r= \sqrt{(x-h)^2 + (y-k)^2}\. Substituting the known values, we get \[ r = \sqrt{(6-2)^2 + (4-1)^2} = \sqrt{16 + 9} = 5\]

Step 3: Write down the equation of the circle

The standard form of the equation is \((x-h)^2 + (y-k)^2 = r^2\). Substituting \(h=2\), \(k=1\), and \(r=5\) into this, we get \[(x-2)^2 + (y-1)^2 = 25\]. This is the equation of the given circle.

Key Concepts

Distance Formula

The distance formula is a great tool in geometry to find how far apart two points are on a coordinate plane. It is derived from the Pythagorean Theorem. The formula is expressed as

• \(\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)

where \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points. To apply it, substitute the coordinates into the formula and simplify.

When dealing with circles, this formula helps us calculate the radius when the center and a point on the circle are known. By plugging in these points, you find the straight line distance between them, which is the radius.

Circle Center

The center of a circle is a vital part of defining its shape. It's simply the point right in the middle of the circle, equally distant from all points on the circle. When you have a circle on a graph, its center often has coordinates \((h, k)\).

This point is used as a reference for other calculations, like finding the radius or writing the equation of the circle. In our given exercise, the circle's center is the point \((2, 1)\). This means all calculations around this circle are based with respect to this reference point.

Remember, knowing the center allows for straightforward swap into circle equations and understanding circle symmetry.

Radius Calculation

Calculating the radius of a circle is straightforward with known points. The radius is the distance between the circle's center and any point on its circumference. As we talked about before, the distance formula comes handy here. Given the center at \((2, 1)\) and the point on the circle \((6, 4)\), simply apply:

• Radius \( r = \sqrt{(6 - 2)^2 + (4 - 1)^2} \)

This simplifies to \( r = \sqrt{16 + 9} = \sqrt{25} = 5 \).

The calculation shows the radius is 5 units long. Knowing this length is critical for accurately plotting the circle on a graph and for forming the circle's equation.

Standard Form of Circle Equation

The standard form of the equation of a circle is a mathematical expression that precisely describes all points on the circle. It is typically written as:

• \((x - h)^2 + (y - k)^2 = r^2\)

Here:

• \((h, k)\) specifies the circle's center
• \(r\) stands for the radius of the circle

In our case, by inserting \(h=2\), \(k=1\), and \(r=5\), we form:

• \((x-2)^2 + (y-1)^2 = 25\)

This representation allows anyone to draw the circle on a coordinate plane by understanding its position and size.

The usefulness of the standard form is that it is very readable, making it simple to identify each element of the circle.