Solve each problem. To visualize the situation, use graph...

Solve each problem. To visualize the situation, use graph paper and a pair of compasses to carefully draw the graphs of the circles. The locations of three receiving stations and the distances to the epicenter of an earthquake are contained in the following three equations: $(x-2)^{2}+(y-4)^{2}=25$ $(x-1)^{2}+(y+3)^{2}=25,$ and $(x+3)^{2}+(y+6)^{2}=100 .$ Determine the location of the epicenter.

Solve each problem. To visualize the situation, use graph paper and a pair of compasses to carefully draw the graphs of the circles.
The locations of three receiving stations and the distances to the epicenter of an earthquake are contained in the following three equations: $(x-2)^{2}+(y-4)^{2}=25$ $(x-1)^{2}+(y+3)^{2}=25,$ and $(x+3)^{2}+(y+6)^{2}=100 .$ Determine the location of the epicenter.

Key Concepts

Circle Equation

A circle equation in standard form, expressed as (x - h)^2 + (y - k)^2 = r^2, represents all points (x, y) that are at a fixed distance r from a central point (h, k). This concept is fundamental since it translates geometrical information about positions and distances into an algebraic form, allowing us to work with circles analytically.

Graphing Circles

Graphing circles involves plotting the center and drawing an accurate circle using a given radius. This visualization aids in understanding the spatial relationships among different circles, particularly in problems that require finding their intersections.

Intersection of Circles

The intersection of circles refers to the point or points where two or more circles meet. When solving problems that require finding a common point among circles, such as identifying an epicenter, determining these intersections is key.

Systems of Equations

A system of equations is a collection of two or more equations with a common solution set. In the context of circle equations, solving a system allows us to find the points that satisfy all conditions simultaneously, representing the intersection points of the circles.

Distance Formula

The distance formula, derived from the Pythagorean theorem, calculates the distance between two points in the coordinate plane. It underpins the circle equation by relating the coordinates of any point on the circle to the center of the circle and the given radius.

Transcript

00:03 Okay, to find the epicenter of the earthquake, we want to find the point of intersection of the three circles.

00:09 So i'm going to start by sketching the circles.

00:11 And if you have compass and graph paper, that's even better for accuracy.

00:17 So the first circle is going to have center to four and radius five.

00:22 So we can sketch that.

00:24 Plot the center and then from there, up five, right five, down five, left five.

00:30 And we get our sketch.

00:34 The second circle has center 1, negative 3, and radius 5.

00:39 So we can plot that point, 1, negative 3, and then up 5, right 5, down 5, left 5.

00:49 We have our circle.

00:52 And then the third one has center negative 3, negative 6, and radius 10.

00:57 So negative 3, negative 6.

00:59 And then we're going to go 10 and write 10.

01:06 I'm probably not gonna be able to make the whole circle, but we get something like that.

01:13 And so we're looking for where these all intersect and it's likely to be somewhere around here.

01:19 So to find that point with algebra, what we're going to do is take our three equations, change their form a little bit, and then solve the system.

01:26 So i'm going to multiply the equation out and get x squared minus 4x plus 4, plus y squared minus 8y plus 16 equals 25 and when i combine some like terms and simplify that i get x squared minus 4x plus y squared minus 8y equals 5 for the second equation multiplying it out gives us x squared minus 2x plus 1 plus y squared plus 6y plus 9 equals 25 and that simplifies to be x squared minus 2x plus y squared plus 6y equals 15.

02:10 And for the third one, multiplying it out gives us x squared plus 6x plus 9 plus y squared plus 12y plus 36 equals 100...

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