Problem 7 - 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. $$ P(-1,5), C(-1,1) $$

Step-by-Step Solution

Answer

The circle's equation is \((x + 1)^2 + (y - 1)^2 = 16\).

Step 1: Calculate the Radius

To find the radius of the circle, use the distance formula between the center \( C(-1, 1) \) and the point \( P(-1, 5) \). The distance formula is \( d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Plug in the coordinates: \( d = \sqrt{((-1) - (-1))^2 + (5 - 1)^2} = \sqrt{0 + 16} = 4 \). So, the radius is 4.

Step 2: Write the Circle Equation in Center-Radius Form

The equation of a circle in center-radius form is \( (x - h)^2 + (y - k)^2 = r^2 \), where \( (h, k) \) is the center and \( r \) is the radius. With center \( C(-1, 1) \) and radius 4, the equation is \( (x + 1)^2 + (y - 1)^2 = 16 \).

Step 3: Write the Circle Equation in Standard Form

The standard form of a circle is the same as the center-radius form, hence it is \( (x + 1)^2 + (y - 1)^2 = 16 \). Both center-radius and standard forms represent the same equation here.

Key Concepts

Understanding the Center-Radius Form

The center-radius form allows us to express the equation of a circle in a clear way. It's based on the position of the center of the circle and the length of its radius. The formula is structured as \((x - h)^2 + (y - k)^2 = r^2\), where

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

This formula directly shows the circle's key attributes, making it easy to see and use. To form the equation of a circle, like in our exercise, we first find the center (by knowing the given coordinates of the center) and the radius (by using points on the circle). This form is visually intuitive and greatly aids in sketching a graph or solving geometrical problems regarding circles. By knowing where the circle is centered and how far it goes out, we can essentially recreate the exact circle just from this equation.

Decoding the Standard Form

The standard form of a circle's equation is often considered synonymous with the center-radius form since it uses the same mathematical expression. Typically, we see this as \((x - h)^2 + (y - k)^2 = r^2\).

• This places emphasis on both the circle's center, \((h, k)\), and its radius, \(r\),

which are already defined within the equation. For practical purposes, this form is simple to use and transforms easily back to give clear values for the circle's attributes. In our specific exercise, the task was to write the equation both in center-radius and standard forms. Interestingly, the formats result in identical expressions due to their inherent nature. Thus, the standard form and center-radius equation provide dual perspectives of the same geometric shape, both being ideal for different scenarios.

Applying the Distance Formula

The distance formula is a handy tool in geometry used to calculate the length between two points in a coordinate plane. Often used in the context of circles, it helps determine the radius when the center and a point on the circle are known. The formula is rightly expressed as \[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

• \((x_1, y_1)\) is one point,
• \((x_2, y_2)\) is the other point,

For our circle example, we used this formula to compute the radius, employing the center \((C(-1, 1))\) and the point \((P(-1, 5))\). We found:

• The x-coordinates are the same,
• only the y-values change, resulting in a straightforward calculation,
• yielding a radius of 4 units.

The distance formula precisely quantifies how to compute the radius, which is foundational to writing the circle's equation. Once we know this, accurately describing the circle in its equation form becomes a seamless task.