Write each equation in standard form to find the center and radius of the circle. Then sketch th

Write each equation in standard form to find the center and...

Key Concepts

Graphing a Circle

Graphing a circle involves plotting its center on the coordinate plane and then drawing a curve that is equidistant from the center at all points, using the radius to determine the distance. This graphical representation helps visualize the circle's properties and its relationship to other geometric elements in the plane.

Identifying the Center and Radius

Once the equation of a circle is written in standard form, the center (h, k) is obtained from the shifts in the x and y terms, while the radius is determined by taking the square root of the constant on the right-hand side of the equation. This step is crucial for understanding and representing the circle's position and size.

Completing the Square

Completing the square is an algebraic technique used to transform a quadratic equation into a perfect square trinomial plus a constant. This method is essential in rewriting the general form of the circle's equation into its standard form, which then makes it possible to easily identify the center and radius.

Standard Form of a Circle

The standard form of a circle's equation is expressed as (x - h)² + (y - k)² = r², where (h, k) represents the center of the circle and r is its radius. This form is fundamental because it directly reveals the geometric properties of the circle and serves as a basis for graphing and analyzing circles in coordinate geometry.

Transcript

00:01 The general equation of a circle in standard form is x minus h squared plus y minus k squared equals r squared.

00:09 Now sometimes we are given the equation of a circle in general form, and we want to know how to get from general form to standard form.

00:17 So take for example, if we are given the equation x squared plus y squared plus 6x plus 4y plus 12 equals 0.

00:31 We want to know what this equation is in standard form.

00:35 So we begin by grouping together our x's and our ys.

00:41 So we have x squared plus 6x and plus another c term plus y squared plus 4y, this another c term equals.

00:55 Now we have plus 12 equals zero in our original equation, but i'm going to go ahead and subtract 12 from both sides, so equals negative 12.

01:06 Now, we want to find these c terms so that we can get a polynomial that will factor into a perfect square so that we can follow our standard form.

01:20 Now, the general equation, or the general form of a polynomial is ax squared plus bx, plus c equals 0.

01:29 So we have ax squared and we have bx, and we need the c.

01:32 So the c is given by b over 2 quantity squared.

01:41 So applying this, we have x squared plus 6x, 6 is our b term.

01:50 So 6x plus 6 over 2 on v squared plus y squared plus 4 y plus now 4 is our b.

02:04 So 4 over 2 quantity squared equals the negative 12.

02:09 Now we added this number and this number are two c terms to the left side.

02:15 We also have to add them to the right side so that our equality holds...