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 Circles

Graphing circles involves plotting the center on the coordinate plane and drawing a circle with a fixed distance (the radius) from that center. This process provides a visual representation of the circle and helps in understanding its spatial relationship with other elements in the coordinate system.

Standard Form of a Circle Equation

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 the radius. This format makes it straightforward to identify and graph the circle by clearly showing its geometric properties.

Completing the Square

Completing the square is a method used to transform a quadratic equation into a form that contains a perfect square trinomial. In the context of circle equations, this technique enables the conversion of the general equation into the standard form, facilitating the identification of the circle’s center and radius.

Transcript

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

00:08 Now, sometimes we're given the equation of a circle in general form, and we want to know how to get those equations in general form to standard form.

00:18 So take for example, if we have the equation x squared plus y squared minus 22y minus 5 equals 0.

00:29 So this is in general form and we are going to convert this into standard form.

00:40 So our first step would be to group together our x terms and our y terms while moving our constant term to the right side of the equation.

00:50 So doing that we get x squared plus y squared minus 22y equals 5.

01:03 Now, looking at our equation above, we know that this is one squared term and then a squared y term.

01:13 So we want this term to be a perfect square polynomial so that it can be factored into the form that we need above.

01:23 So to do that, we're going to have to complete the so the general form of a polynomial is a x squared plus bx plus c.

01:36 So we have a x squared and we have bx and we are looking for this c value.

01:42 Now c is b over 2 squared.

01:48 So doing this gives us x squared plus i squared minus 22 y plus 2 now our b term is negative 22.

02:01 So negative 22 over 2 squared equals 5.

02:07 And now we added a term to the left side.

02:10 So we also have to add this term to the right side so that our equality still holds.

02:14 So 5 plus negative 22 over 2 squared.

02:20 So simplifying this equation, we get x squared plus y squared minus 22 y plus 2 plus and negative 22 over 2 is 11, and negative 11 squared is 121.

02:37 So equals 5 plus 121 is 126.

02:42 Now looking at this polynomial in terms of y, we want to factor this...

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