Circle Problem

Question

Consider the circle that passes through the given points below. Consider the chords PQ and QR. Find the equations for the perpendicular bisectors of these chords. Since the perpendicular bisectors of any 2 chords of a circle must intersect in the center of the circle, use these 2 equations to find the center of the circle. Find the radius of the circle. Use your center and radius to find the equation of the circle in graphing form, (x-h)^2+(y-k)^2=r^2. Then, multiply the graphing form of your answer out so that your final answer is in the form x^2 +y^2+ax +by +c=0. P(-1,3), Q(-1,-2), R(0,-4)

Answer 1

The chord PQ is a vertical line segment.  So its perpendicular bisector would have a slope of 0 and a y-intercept equal to its midpoint (3 - 2)/2 which is 1/2.  For QR its slope is -2/1 and its midpoint is (-1+0)/2 and (-2-4)/2 which is (-1/2, -3).  so the equations of the perpendicular bisectors are

 

y = 1/2

y = x/2 - 11/4

 

Now solve for x to find the center of the circle

 

1/2 = x/2 - 11/4

13/4 = x/2

4x = 26

x = 6.5

 

So the center of the circle is (6.5, 0.5) and the equation is (x-6.5)²+(y-0.5)²=r².  Find r by substituting in one of the points

 

(-1 - 6.5)² + (3 - 0.5)² = r²

(-7.5)² + (2.5)² = r²

56.25 + 6.25 = r²

r = √62.5 = 7.9

 

So the equation is (x - 6.5)² + (y - 0.5)² = 7.9².  I'll let you multiply it out.