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If $A = (3,4)$ and $C = (-7,2)$ are opposite vertices of a rectangle $ABCD,$ then vertices $B$ and $D$ must lie on the circle
\[x^2 + y^2 - px - qy + s = 0.\]
Compute the ordered triple of real numbers $(p,q,s).$

 Aug 14, 2023
 #1
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B= (-7,4)      C = (3,2)

 

The midoint of BD =  M =   [ ( -7 + 3) / 2 , (2 + 4) /2] =  (-2, 3)  = the center of the  circle

 

The  distance   from  B to M =   sqrt [ (-7 - -2)^2 + (4 -3)^2 ] =  sqrt [ 25 + 1] = sqrt (26) = radius of the  circle

 

So we have

 

(x - -2)^2  + ( y - 3)^2 = 26

 

( x + 2)^2  + ( y -3)^2 = 26

 

x^2 + 4x + 4 + y^2 - 6y + 9 = 26

 

x^2 + y^2 + 4x - 6y - 13 =  0

 

x^2 + y^2 - (-4)x - 6y + (-13)  = 0

 

(p , q, s) = (-4, 6, -13)

 

 

cool cool cool

 Aug 14, 2023

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