Problem 73 - Algebra 2

Question

Write an equation of a circle with the given center and radius. center \((2,-3),\) radius 6

Step-by-Step Solution

Answer

The equation of the circle with the given center and radius is: \((x - 2)^2 + (y + 3)^2 = 36\).

Step 1: Identify the given values

Identify the given center and radius of the circle. Center is given as \((2,-3)\), so the values of \(h\) and \(k\) are 2 and -3 respectively. Radius is given as 6, so the value of \(r\) is 6.

Step 2: Substitute values into the standard form equation

Plugging these values to our standard circle equation we get: \((x - 2)^2 + (y - (-3))^2 = 6^2\).

Step 3: Simplify the equation

Simplify the equation to its final form: \((x - 2)^2 + (y + 3)^2 = 36\).

Key Concepts

Standard Form Equation

The equation of a circle in the standard form is a key concept in coordinate geometry. It is expressed as \[(x - h)^2 + (y - k)^2 = r^2,\]where

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

To write an equation for a circle, you simply plug the center coordinates and radius into this equation.
For example, in the exercise, the center was given as \((2, -3)\) and the radius as \(6\). Substituting these into the standard form equation: \((x - 2)^2 + (y + 3)^2 = 36\). Here, the \(36\) is the radius squared \(6^2\). This format is often the easiest way to write and understand a circle's equation because it directly reflects the circle's geometrical properties.

Center-Radius Form

The center-radius form of a circle's equation is essentially the same as the standard form. However, it's emphasized as highlighting the circle's core properties:

• The center, \((h, k),\) gives you the exact point around which the circle is drawn.
• The radius, \(r,\) defines how far the boundary of the circle is from its center.

When you have the center and radius, you have everything you need to describe a circle precisely.
This form makes it easy to visualize the circle and understand its location and size on a coordinate plane. For the exercise, using the center-radius form helped us quickly identify the circle's position and accurately plug the values \((h = 2, k = -3, r = 6)\) into our equation.

Coordinate Geometry

In coordinate geometry, circles are beautifully symmetrical shapes, described mathematically to determine their location exactly on a plane. It involves using coordinates to express the geometric properties of shapes. For circles, you need:

• The circle's center point, which provides the base coordinates for positioning.
• The radius, which denotes the circle's size and ensures all points on the circle are equidistant from the center.

Using the example exercise, the circle encompassed using coordinate geometry is centered at \((2, -3)\) with each point 6 units away from it. This exercise illustrates how coordinate geometry uses equations to describe spatial properties of figures, enabling precise calculations and graphical representations.