Problem 4 - 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. $$ P(0,-3), C(0,0) $$

Step-by-Step Solution

Answer

The equation of the circle is \( x^2 + y^2 = 9 \), both in center-radius and standard form.

Step 1: Identify Given Information

We are given that the point on the circle is \( P(0, -3) \) and the center of the circle is at \( C(0, 0) \).

Step 2: Compute the Radius

The radius of the circle is the distance between the center \( C(0, 0) \) and point \( P(0, -3) \). Using the distance formula: \( r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \). Here, this becomes \( r = \sqrt{(0 - 0)^2 + ((-3) - 0)^2} = \sqrt{0 + 9} = 3 \). So, the radius \( r = 3 \).

Step 3: Equation in Center-Radius Form

The center-radius form for the equation of a circle is \( (x - h)^2 + (y - k)^2 = r^2 \). With \( h = 0 \), \( k = 0 \), and \( r = 3 \), the equation becomes \( (x - 0)^2 + (y - 0)^2 = 3^2 \), which simplifies to \( x^2 + y^2 = 9 \).

Step 4: Equation in Standard Form

The standard form of the equation of a circle is the same as the center-radius form provided the center is at the origin: \( x^2 + y^2 = 9 \). Since the center is at \( (0,0) \), the equations are identical.

Key Concepts

Center-Radius Form

In mathematics, the center-radius form of a circle is a very intuitive way of understanding how a circle is defined. This form is focused on highlighting the circle's center point and its radius, which is the distance from the center to any point on the circle's edge. The equation in center-radius form is written as: \[ (x - h)^2 + (y - k)^2 = r^2 \] Here:

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

To find the radius, one would typically first be given or would calculate the distance between the center and a point on the circle. For instance, if a circle's center is at \( (0, 0) \) and a point on the circle is \( (0, -3) \), the radius \( r \) would be calculated using the distance formula, resulting in a radius of 3.With this information, substituting \( h = 0 \), \( k = 0 \), and \( r = 3 \) into the equation, we get \[ (x - 0)^2 + (y - 0)^2 = 3^2 \] which simplifies to \[ x^2 + y^2 = 9 \] This equation clearly represents the circle's fundamental features: its center and its radius squared.

Distance Formula

The distance formula is a fundamental concept used not only in geometry but in various fields requiring measurement of space between two points. It's derived from the Pythagorean theorem and is extremely useful for calculating distances in the coordinate plane. The formula is expressed as:\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] Where:

• \((x_1, y_1)\) and \((x_2, y_2)\) are the coordinates of the two points.
• \(d\) is the distance between these points.

Applying this formula to find the radius of a circle involves using the circle's center as one point and a point on the circumference as the other. For example, if a circle's center is at \(C(0, 0)\) and a point on the circle is \(P(0, -3)\), substituting into the formula provides:\[ r = \sqrt{(0 - 0)^2 + ((-3) - 0)^2} \] After simplifying, this gives us \[ r = \sqrt{9} = 3 \] as the circle's radius. This distance is crucial for determining the center-radius form of a circle's equation.

Standard Form of a Circle

The standard form of a circle's equation is a specific notation used so frequently that it becomes second nature in solving geometrical problems. When the circle is centered at the origin, this form is wonderfully simple and is represented as: \[ x^2 + y^2 = r^2 \] When a circle's center isn't at the origin, the equation takes the shape used in the center-radius form, but simplified to standard form. Particularly, when the center is at the origin like \(h = 0 \) and \(k = 0\), it maintains a simple quadratic relation. This equation naturally extends the Pythagorean theorem across all radii of the circle because each point \( (x, y) \) on the circle maintains the condition that the distance from the center equals the radius.Consider the circle from our example with a center at \( (0,0) \) and radius of 3;The equation would thus be:\[ x^2 + y^2 = 3^2 \]This shows that every point \((x, y)\) on this circle is exactly 3 units away from the center, seamlessly illustrating the primary characteristic of circles in this form.