Question 1037037: Point B(-2, 4) lies on a circle centered at A(1, 3). Write a paragraph proof to determine whether C(4, 2) also lies on the circle.
Found 2 solutions by josgarithmetic, Aldorozos:
Answer by josgarithmetic(39618)   (Show Source):
You can put this solution on YOUR website! Find the equation using standard form as reference and use of the Distance Formula. Your paragraph will be based on this, and also on the use of your point C in your equation, that point C should (which you need to check) satisfy your equation.
To push everything together to make the unrefined equation,
 .
Simplify this and check to find if (4,2) satisfies the equation.
Answer by Aldorozos(172) (Show Source):
You can put this solution on YOUR website! There are several ways to solve this problem.
We know that the equation of a circle is (x-xo)^2+(y-yo)^2 = C^2
where xo and yo are the coordinates of the circle. In this case A(1,3)
Therefore the equation of the circle is (x-1)^2+(y-3)^2 = c^2
Now we have to find C. C is the radius. The radius is the distance between the center of the circle to one of the points on the circle. We already have one point (-2,4) and we already have the coordinates of the center (1,3). When we have to points we can calculate the distance between the points by using the distance formula = radius = c = square root of (y2-y1)^2+(x2-x1)^2. We are looking C^2. Therefore instead of calculating the square root we calculate (y2-y1)^2+(x2-x1)^2 which is equivalent to C^2 = Radius^2 = (4-2)^2+(2-3)^2 = 10
Therefore the equation of this circle is (x-1)^2+(y-3)^2 = 10
Any point on the circle has to satisfy the equation (x-1)^2+(y-3)^2 = 10
We replace x and y with the given point (4,2). If we replace x with 4 and you with 2 we get (4-1)^2 + (2-3)^2 = 9+1 = 10. Because both sides of the equation are equal this means that the point (4,2) is on the circle. Any point that is not on the circle won't satisfy the equation. Only points on the circle will give us the result of 10 = 10
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