Consider the equations x^2+y^2=16 and (x-2)^2+(y-2)^2=16. By...

Key Concepts

Intersection Conditions of Circles

To check whether two circles intersect, one compares the distance between their centers with the sum and the absolute difference of their radii. Specifically, if the distance is less than the sum of the radii and greater than the absolute difference, the circles intersect at two points. This concept is key in solving problems involving circle intersections.

Circle Equation

The circle equation in the Cartesian plane is given in the form (x - h)^2 + (y - k)^2 = r^2, where (h, k) represents the center of the circle and r is its radius. This concept is fundamental in understanding the geometric properties of circles and how they are graphically represented.

Center and Radius

Extracting the center and radius from the equation is essential for analyzing the circle's position and size. The center indicates the circle's location in the plane, while the radius measures the distance from the center to any point on the circle, which is crucial for determining potential intersections with other circles.

Distance Formula

The distance formula, derived from the Pythagorean theorem, computes the distance between two points in the Cartesian plane. When analyzing circles, calculating the distance between their centers helps determine how the circles are placed relative to each other and whether they might intersect.

Transcript

00:02 Hello.

00:03 So the equation we're given here is 2x squared plus 2y squared minus 16 is equal to 4x.

00:10 Now we know that the standard equation of a circle centered at the point h comma k with radius r is equal to x minus 8 squared plus y minus k squared is equal to r squared.

00:22 So how can we go ahead and take this equation and put it into this form? right now, it's not in this form.

00:28 We don't have, you know, some quantity squared or some quantity squared is equal to a constant.

00:32 So let's go ahead.

00:34 First, we have an x term on the right hand side of the equation.

00:38 Let's move that to the other side and move the consonant to the right side.

00:42 So then we have 2x squared plus 2y squared and then minus 4x is then equal to, well, 16.

00:53 So now let's first, let's go ahead.

00:57 Let's divide everything through here by 2, right? because we have the coefficients on the, well, so if you just had x squared, that would be like an x minus zero squared.

01:06 We had this coefficient of 2 in front.

01:09 Let's go ahead and divide everything through by 2.

01:12 So doing so, that would give us x squared plus y squared minus 2x is equal to, well, 16, divided by 2 is 8.

01:26 Now, to get ourselves a perfect, we have this negative 2x term.

01:31 We just had x squared plus y squared is equal.

01:34 Of 8, well, that would be a circle with radius square root of 8.

01:41 That would be x minus 0 squared plus y minus 0 squared is equal to r squared.

01:45 Well, in that case, r would be the square root of 8.

01:47 But we have this negative 2x term.

01:50 So let's put our x terms together.

01:52 We have the x squared and then we have minus 2x.

01:56 Now, what we want to do is go ahead and create a perfect square here by completing the square.