Question 530394: I'm so lost on this equation:
Find the equation of the circle centered at (4,6) passing (2,2).
Thank you for the person that is willing to help!
Answer by lmeeks54(111)   (Show Source):
You can put this solution on YOUR website! The standard form of the equation for a circle is:
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(x - h)^2 + (y + k)^2 = r^2
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Where h = the x coordinate of the center of the circle
Where k = the y coordinate of the center of the circle
Where r = the radius of the circle
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We are given the center of the circle: (4, 6)
We are given one of the points the circle passes through: (2, 2)
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The only thing we are missing to complete the equation for the circle is to know (if given) or derive (if not given), r, the radius of the circle.
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With the two points given: center = 4, 6 and circle passes through 2, 2; if we connect those two points together, that line = the radius of the circle.
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To determine the radius, think about a right triangle that has as its "a" side and "b" side the rise and run difference between the two given points. And, knowing the Pythagorean theorem to determine the hypotenuse of a right triangle:
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a^2 + b^2 = c^2
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We know that in in this instance the hypotenuse = the radius of our circle, so we will substitute r for c. If we take the square root of both sides of the previous equation we get:
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r = SQRT(a^2 + b^2)
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To solve:
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a = the difference in the two x coordinates (from the two points given):
a = 4 - 2 = 2
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b = the difference in the two y coordinates (from the two points given):
b = 6 - 2 = 4
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r = Sqrt(2^2 + 4^2)
r = Sqrt(4 + 16)
r = Sqrt(20)
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To return to the original problem: determine the equation for the circle with the given center point and passing through the given point:
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(x - 4)^2 + (y + 6)^2 = Sqrt(20)^2
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which could be simplified to:
(x - 4)^2 + (y + 6)^2 = 20
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cheers,
Lee
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