Problem 50 - Algebra 2

Question

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

Step-by-Step Solution

Answer

The equation of the circle with center \((6,4)\) and passing through point \((2,1)\) is \((x-6)^2 + (y-4)^2 = 25\)

Step 1: Calculate the radius

Since the radius can be found using the distance formula \(r = \sqrt{(x-h)^2 + (y-k)^2}\), where \(x\), \(y\) is a point on the circle and \((h, k)\) is the center of the circle. Substituting \(x=2\), \(y=1\), \(h=6\), \(k=4\) into the formula will provide \(r = \sqrt{(2-6)^2 + (1-4)^2} = \sqrt{(16+9)} = 5\)

Step 2: Formulate the equation of the circle

Now, substitute the values of \(h\), \(k\) and \(r\) into the general equation of a circle \((x-h)^2 + (y-k)^2 = r^2\). This will lead to: \((x-6)^2 + (y-4)^2 = 5^2\)

Step 3: Simplify the equation

As the last step, simplify the equation of the circle by expanding the square terms and collecting like terms. This gives the final equation of the circle as \((x-6)^2 + (y-4)^2 = 25\)

Key Concepts

Distance Formula

The distance formula is an essential tool in coordinate geometry to determine the length between two points. It originates from the Pythagorean Theorem. When given two points \( (x_1, y_1) \) and \( (x_2, y_2) \), you can find the distance between them using the formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\]Here, \(d\) represents the distance between the points. This formula is crucial for finding the radius of a circle when you know a point on the circle and its center. In the specific example of a circle with center \( (6,4) \) that goes through \( (2,1) \), we apply the distance formula:

• Calculate \( (2-6)^2 = 16 \)
• Calculate \( (1-4)^2 = 9 \)
• Sum them up: \( 16 + 9 = 25 \)
• Then, take the square root: \( \sqrt{25} = 5 \)

Thus, the radius is 5 units long.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves placing geometric figures within a coordinate system. Points are defined using ordered pairs \( (x, y) \), allowing you to explore relationships and measurements.

Circles in Coordinate Geometry

In this concept, a circle's position on the coordinate plane is a center point \( (h, k) \) with a radius \( r \). The general equation for any circle is given by:\[(x-h)^2 + (y-k)^2 = r^2\]This equation ensures every point \( (x, y) \) on the circle's boundary is the same distance \( r \) from the center \( (h, k) \). If a circle's center is known, and you have a point on the circle, you can determine the radius and write its equation.

Radius Calculation

Calculating a circle's radius when given a center and a point on the perimeter requires using the distance formula. The radius is simply the distance from the center to the given point on the circle.

Why Radius Matters

The radius is a critical component of a circle's equation. It determines the overall size and positioning of the circle relative to its center. By substituting the values of the center point \( (h, k) \) and a point \( (x, y) \) on the circle into the distance formula:\[r = \sqrt{(x-h)^2 + (y-k)^2}\]You can accurately calculate the length of the radius. For example, with the point \( (2,1) \) and center \( (6,4) \), the radius comes out as 5 units. This measurement becomes vital for forming the circle's equation, as seen in the problem definition.