What is the equation of the circle?
a circle has an endpoint of a diameter at (-2,4).if the diameter is perpendicular to a chord (which is not a diameter) whose endpoints are (6,8) and (
a circle has an endpoint of a diameter at (-2,4).if the diameter is perpendicular to a chord (which is not a diameter) whose endpoints are (6,8) and (6,0) find the equation of the circle.
or:a circle has an endpoint of a diameter at (-2,4).if the diameter is perpendicular to a chord (which is not a diameter) whose endpoints are (6,8) and (6,0) find the equation of the circle.
or:Consider the three given pointsA( -2, 4), B( 6, 0), C( 6, 8) as the verteces of a triangle, and a = BC, b = AC, c = ABas its sides, which are chords of the circumcircle of this triangle. The centre point M of the circumcircle is defined as the intersection point of the triangle's perpendicular side bisectors.One such bisector is already given with the condition: \"if the diameter is perpendicular to a chord (which is not a diameter) whose endpoints are B(6,8) and C(6,0)\".A diameter by definition passes through the centre point M, and the perpendicular bisector of each possible chord passes through M. As a=BC is vertical ( xB = xC = 6 ), the perpendicular bisector of side a, a', has to be horizontal. As it goes through A(-2,4) it has to bisect side a in Pa( 6, 4). Therefore, yM = 4, and a' := y = 4.Now we have to find the bisecting point Pc of one of the other sides, say, side c:Pc( xB - (( xB - xA) / 2 ), yB - (( yB - yA) / 2 ) )Insert the values:Pc( 6 - ((6 - (-2)) / 2 ), 0 - ((0 - 4 ) / 2) )Pc( 2, 2).Next we have to find the slope mc of c:mc = ( yB - yA) / ( xB - xA)Insert the values and you get:mc = 1/2The slope of the perpendicular mc' is the negative inverse of mc. Hence,mc' = -2.Now we have a point and a slope, so we can build the linear equation in point slope form, in order to find xM by equating it with a' := y = 4 :c' := (y - yPc) = mc(x - xPc)y - 2 = 2 (x - 2)y = 2x - 2a' = c' 4 = 2x - 2x = 3.We have determined M( 3, 4).The equation of a circle is :(x - xM)^2 + (y - yM)^2 = r^2Let's insert, say, point C( 6, 8) (and M, of course):(6 - 3)^2 + (8 - 4)^2 = r^29 + 16 = 25r = 5.The equation of your circle is then, finally:( x - 3 )\u00b2 + ( y - 4 )\u00b2 = 25.
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