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anonymous
 one year ago
Write the general equation for the circle that passes through the points:
(1, 7)
(8, 6)
(7, 1)
You must include the appropriate sign (+ or ) in your answer.
x^2 + y^2__x__y=0
anonymous
 one year ago
Write the general equation for the circle that passes through the points: (1, 7) (8, 6) (7, 1) You must include the appropriate sign (+ or ) in your answer. x^2 + y^2__x__y=0

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mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1The equation of a circle with center (x, y)and radius r is \((x  h)^2 + (y  k)^2 = r^2\) You have three given points. Substitute one point at a time into the equation above to get 3 three equations in three unknowns, h, k, and r. Solve the system of equations and substitute the values of h, k and r in the above equation.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Ok so I have to put (1,7) into that equation you gave me??

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1That is one way of doing it but it's a lot of work unless you have an electronic way of solving the system of equations.

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1Here is another way of solving.

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1Here is a circle with two points on it and a chord drawn through them. dw:1435785536910:dw

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1dw:1435785591709:dw

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1The perpendicular bisector of the chord passes through the center of the circle. dw:1435785640579:dw

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1You are given three points on the circle. dw:1435785694244:dw

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1Use a chord again and a perpendicular bisector again: dw:1435785724236:dw

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Ok so how can I figure out the equation by using that method?

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1Where the perpendicular bisectors of the chords intersect is the center of the circle.

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1Ok. Let's start. dw:1435785943459:dw

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1dw:1435786061026:dw

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1dw:1435786091850:dw

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1Let's start with points (1, 7) and (8, 6) We need to find the equation of the perpendicular bisector of the segment with those endpoints.

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.11. Find the slope of the segment \(m = \dfrac{6  7}{8  1} = \dfrac{1}{7} = \dfrac{1}{7} \)

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1The slope of the perpendicular bisector is m = 7

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1The perpendicular bisector passes through the midpoint of the segment, so we need to find the midpoint of the segment.

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1midpoint = \((\dfrac{1 + 8}{2}, \dfrac{7 + 6}{2}) = (\dfrac{9}{2}, \dfrac{13}{2})\) dw:1435786426317:dw

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1We have a point and the slope, so we can find the equation of that perpendicular bisector.

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1\(y = mx + b\) \(\dfrac{13}{2} = 7(\dfrac{9}{2}) + b\) \(\dfrac{13}{2} = \dfrac{63}{2} + b\) \(b = 25\) \(y = 7x  25\) This is the equation of the first perpendicular bisector of a chord.

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1Now we need to choose another set of two points on the circle and do the same.

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0Ok so how would I plug that in this equation? x^2 + y^2__x__y=0

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1dw:1435786849703:dw

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1dw:1435786887498:dw

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1Now we have the chord in black in the last figure above. We need its slope and its midpoint.

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1\(slope = m = \dfrac{1  7}{7  1} = \dfrac{8}{6} = \dfrac{4}{3}\) The slope of the perpendicular bisector is \(\dfrac{3}{4} \)

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1We need the coordinates of the midpoint of the chord. midpoint \(= (\dfrac{7 + 1}{2}, \dfrac{1 + 7}{2}) = (4, 3)\)

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1Since we now have the slope of the perpendicular bisector and a point it goes through, we can find the equation of the perpendicular bisector.

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1\(y = mx + b\) \(3 = \dfrac{3}{4} \times 4 + b\) \(b = 0\) The eq of the perp bisector is: \(y = \dfrac{3}{4} x \)

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1Now we have two equations of the two perpendicular bisectors of chords that we know intersect at the center pf the circle. We solve the system of equations: \(y = 7x  25\) \(y = \dfrac{3}{4} x\) Since both equations are solved for y, we can equate the right sides and solve for x. \(7x  25 = \dfrac{3}{4}x \) \(28x  100 = 3x\) \(25x = 100\) \(x = 4\) \(y = \dfrac{3}{4} x = \dfrac{3}{4} \times 4 = 3\) Now we know the center is (4, 3).

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1dw:1435787686770:dw

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1We can already fill in a little in the equation of the circle: The center is (h, k), and we know it is (4, 3). \((x  4)^2 + (y  3)^2 = r^2\)

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1All we need now the to find the radius, so we can find r in the equation of the circle.

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1The radius of a circle is the distance between the center of the circle and any point on the circle. We know the center is (4, 3) Also, we were given three points on the circle. We now need to find the distance between the center and any of the three points we were given to find the radius.

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1Let's use point (8, 6) The distance between point (8, 6) on the circle, and point (4,3) the center is the radius of the circle. We use the distance formula: \(d = \sqrt{(x_2  x_1)^2 + (y_2  y_1)^2} \)

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0So it would be 5 right?

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1\(d = \sqrt{(8  4)^2 + (6  3)^2} \) \(d = \sqrt{4^2 + 3^2} \) \(d = \sqrt{16 + 9} \) \(d = \sqrt{25} \) \(d = 5 \)

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1Right. The radius is 5. Since the equation of a circle has r^2 on the right side, we can finish now: \((x  4)^2 + (y  3)^2 = 5^2\) or if you prefer \((x  4)^2 + (y  3)^2 = 25\)

mathstudent55
 one year ago
Best ResponseYou've already chosen the best response.1dw:1435788411933:dw
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