Problem 10 - 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. $$ P(4,2), C(0,1) $$

Step-by-Step Solution

Answer

Center-radius form: \((x - 0)^2 + (y - 1)^2 = 17\); standard form: \(x^2 + (y - 1)^2 = 17\).

Step 1: Understand the Problem

We need to write the equation of a circle given a point on the circle \(P(4,2)\) and its center \(C(0,1)\). We need it in both center-radius form and standard form.

Step 2: Determine the Circle's Radius

The radius \(r\) of the circle can be found by calculating the distance between the center \(C(0,1)\) and the point \(P(4,2)\). Use the distance formula: \(r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\).

Step 3: Apply the Distance Formula

Substitute \((x_1, y_1) = (0, 1)\) and \((x_2, y_2) = (4, 2)\) into the distance formula: \[r = \sqrt{(4 - 0)^2 + (2 - 1)^2} = \sqrt{4^2 + 1^2} = \sqrt{16 + 1} = \sqrt{17}\].

Step 4: Write the Equation in Center-Radius Form

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 \(h = 0\), \(k = 1\), and \(r = \sqrt{17}\), we get \((x - 0)^2 + (y - 1)^2 = (\sqrt{17})^2\).

Step 5: Simplify Center-Radius Equation

Simplify the equation to get \((x - 0)^2 + (y - 1)^2 = 17\). This is the center-radius form of the circle's equation.

Step 6: Convert to Standard Form

The standard form is the same as center-radius form for circles and already given as \((x - 0)^2 + (y - 1)^2 = 17\).

Step 7: Simplify to Standard Form

For clarity, recast the equation explicitly: \(x^2 + (y - 1)^2 = 17\). This is the equation in standard form.

Key Concepts

Distance Formula

The distance formula is a nifty tool in geometry, helping us calculate the distance between two points on the coordinate plane. If you have two points, say \( (x_1, y_1) \) and \( (x_2, y_2) \), the distance formula is given as:

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

This formula is derived from the Pythagorean Theorem. It finds the hypotenuse (or the straight-line distance) in a right triangle, where the differences in x-values and y-values are your legs. For our circle's radius, this is how we calculate the distance from the center to a point on its edge, ensuring it remains consistent across the whole shape.

Center-Radius Form

The center-radius form of a circle's equation is very useful for figuring out a circle's key dimensions at a glance. It's expressed as:

• \( (x - h)^2 + (y - k)^2 = r^2 \)

Here, \( (h, k) \) stands for the circle's center coordinates, and \( r \) is the radius. With this format, you can easily see where the center is and how big the circle stretches. From our problem, \( (h, k) = (0, 1) \), and by plugging in our radius of \( \sqrt{17} \,\), we get \( (x - 0)^2 + (y - 1)^2 = 17 \). This lets anyone quickly understand the circle's layout.

Standard Form

While the center-radius form is intuitive, the standard form of a circle's equation is convenient, especially in various mathematical contexts. It looks like this:

• \( (x - h)^2 + (y - k)^2 = r^2 \)

Interestingly, for circles, the center-radius and standard forms showcase the same equation structure. Thus, the previously derived center-radius form \( (x - 0)^2 + (y - 1)^2 = 17 \) automatically serves as the standard form. This makes moving between forms straightforward for circles, only requiring minor adjustments if ever needed.

Coordinates

Coordinates are a fundamental part of geometry, helping us pinpoint any location on a plane with precision. They are expressed as pairs \( (x, y) \,\), where \( x \) corresponds to the horizontal position and \( y \) to the vertical one. In our circle's context:

• The center is at \( (0, 1) \)
• The point on the circle, \( P \,\) is at \( (4, 2) \)

These coordinates are what we use along with the distance formula to find the radius. They also allow us to express the circle's equation in both the center-radius and standard form, showcasing how essential coordinates are to circle equations.

Radius Calculation

Radius is the consistent distance from the center of a circle to any point on its circumference. To find it in our exercise, we used the distance formula between the center and a point on the circle. Calculating:

• Plug in the center \( (0, 1) \) and point \( (4, 2) \) into the distance formula.
• Solve to find \( r = \sqrt{(4 - 0)^2 + (2 - 1)^2} = \sqrt{17} \).

This calculated distance of \( \sqrt{17} \,\) is not just our radius but a key part of forming both our center-radius and standard form equations. Its calculation ensures that our circle's equation accurately represents its size and position in the coordinate plane.