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

Standard Form of a Circle

The standard form of a circle's equation is (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r is the radius. This form allows for an immediate understanding of the circle's position and size, and is essential in geometry for analyzing and graphing circles.

Completing the Square

Completing the square is an algebraic method used to transform a quadratic equation into a perfect square form. In the context of circles, this technique is employed to rewrite a general quadratic equation into its standard form, enabling the extraction of the circle's center and radius.

Graphing Circles

Graphing circles involves plotting the circle's center on a coordinate plane and drawing a circle with the given radius. Understanding the standard form of a circle is critical because it directly provides the necessary information for accurate sketching by clearly indicating both the center and the distance from the center to any point on the circle.

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.

00:10 Now sometimes we're given the equation of a circle in general form and we need to know how to get between these two forms.

00:16 So take for example if we have the equation x squared plus y squared plus 6x minus 8y minus 6 equals 0.

00:28 So we have this equation of a circle in general form and we want to know what this equation is in standard form.

00:34 So we start by grouping together our x and our y terms.

00:40 So x squared plus 6x plus and then we're going to have another value here that we need to figure out.

00:48 Then plus y squared minus 8y plus another value equals 6.

00:56 Now notice here we have a negative 6 and i will went ahead and added that 6 to the other side of the equation.

01:03 So now we want to know what two values go in these blanks to make these polynomials perfect squares so that they follow the form of our standard general equation.

01:18 So the standard form of a polynomial is a x squared plus bx plus c equals zero.

01:28 So we are looking for the c value.

01:33 Now to find that, our c is equal to b over two quantity squared.

01:40 So we know our b squared here is 6.

01:44 So we know our c is x squared plus 6x plus 6 over 2 squared.

01:53 Then we do the same for our ys.

01:57 So plus y squared minus 8y plus now our b is negative 8, so negative 8 over to quantity squared equals 6.

02:10 Now we added another number, another term to the left side of our equation.

02:15 So we have to do the same to the right side of our equation to make sure that it is still equal, that our quality still holds.

02:23 So you have to add 6 over to quantity squared, and we have to add negative 8 over to quantity d squared to our right side also.

02:34 Now let's simplify this equation...

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