Problem 9 - Algebra 2 and Trigonometry

Question

In \(3-10\) , the coordinates of point \(P\) on the circle with center at \(C\) are given. Write an equation of each circle: a. in center-radius form b. in standard form. $$ P(1,1), C(6,13) $$

Step-by-Step Solution

Answer

The equation of the circle is \((x - 6)^2 + (y - 13)^2 = 169\) in both center-radius and standard form.

Step 1: Identify the Center and a Point on the Circle

We are given that the center of the circle \( C \) is at \((6, 13)\), and point \( P \) on the circle is \((1, 1)\).

Step 2: Calculate the Radius

The radius \( r \) can be found using the distance formula between the center \( C(6, 13) \) and the point \( P(1, 1) \) on the circle. The formula is:\[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]Substituting the coordinates, we get:\[ r = \sqrt{(1 - 6)^2 + (1 - 13)^2} \]\[ r = \sqrt{(-5)^2 + (-12)^2} \]\[ r = \sqrt{25 + 144} \]\[ r = \sqrt{169} \]\[ r = 13 \].

Step 3: Write the Center-Radius Form of the Equation

The center-radius form of the circle's equation is:\[ (x - h)^2 + (y - k)^2 = r^2 \]where \((h, k)\) is the center and \(r\) is the radius. Substituting \((h, k) = (6, 13)\) and \(r = 13\), the equation becomes:\[ (x - 6)^2 + (y - 13)^2 = 169 \].

Step 4: Write the Standard Form of the Circle's Equation

The standard form of a circle's equation is the same as the center-radius form: \[ (x - 6)^2 + (y - 13)^2 = 169 \].Hence, the standard form of the circle's equation is still \[ (x - 6)^2 + (y - 13)^2 = 169 \].

Key Concepts

Center-Radius Form

When dealing with circles, the center-radius form is often the most intuitive way to write an equation. It clearly shows both the center point and the radius, making it easy to visualize the circle. The center-radius form of a circle's equation is: \[(x - h)^2 + (y - k)^2 = r^2\] Where:

• \((h, k)\) is the center of the circle.
• \(r\) is the radius of the circle.

For example, if a circle has a center \(C\) at \((6, 13)\) and a radius of 13, its equation in center-radius form would be \( (x - 6)^2 + (y - 13)^2 = 169 \). This form is very straightforward, as it directly shows you everything you need to know about the circle's size and position.

Standard Form

The standard form of a circle's equation is surprisingly the same as the center-radius form. It is still written as: \[(x - h)^2 + (y - k)^2 = r^2\] While the term 'standard form' might suggest something different or more complex, it really just emphasizes the importance of a structured, recognizable equation.
For example, if a circle has its center at \((6, 13)\) and a radius of 13, the equation in this form would be \((x - 6)^2 + (y - 13)^2 = 169\).
People often use these forms interchangeably, as they essentially convey the same information about a circle. Keeping it in this format helps in consistently recognizing the components of a circle.

Distance Formula

The distance formula is a crucial tool in geometry that helps find the distance between two points on a coordinate plane.
It comes in handy when calculating the radius of a circle when given the center and a point on the circle. The formula is written as: \[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] In this equation:

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
• \(r\) represents the distance between the points, which can be the radius in certain problems.

Using this formula, if point \(P\) is \((1, 1)\) and the center \(C\) is \((6, 13)\), the distance or radius \(r\) would be \(13\), as calculated by substituting into the formula.

Circle Radius Calculation

Finding the radius of a circle is a pivotal step when working with circle equations. Knowing the radius allows one to write both the center-radius and standard forms of a circle’s equation.
When given the center point \((h, k)\) and another point \((x_1, y_1)\) on the circle, the radius can be determined using the distance formula: \[ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Suppose you're given a center at \(C(6, 13)\) and a point on the circle at \(P(1, 1)\).

• First, substitute the points into the distance formula to find the radius: \[ r = \sqrt{(-5)^2 + (-12)^2} \]
• Calculate the squared differences and sum them: \[ 25 + 144 = 169 \]
• The square root of the sum yields the radius: \[ r = \sqrt{169} = 13 \]

With this step, you confirm the circle's size, allowing you to correctly apply and utilize other circle equations.