If each point of the circle \(x^2 + y^2 = 25\) is reflected in the point \((4,1)\), the set of image points satisfies the equation \(x^2 + ay^2 + bx + cy + d = 0\). Compute the ordered quadruple \((a,b,c,d)\) of real numbers.
+6248 \(\text{The center of the original circle is }(0,0)\\ \text{The center of the reflected circle is }(8,2) \text{ (can you see why?)}\\ \text{The radius is unchanged. The equation of the new circle is }\\ (x-8)^2 + (y-2)^2 = 25\\ x^2 - 16x + 64 + y^2 - 4y + 4 = 25\\ x^2 +y^2 -16x-4y+43=0\\ a=1,~b=-16,~c=-4,~d=43\)
.+6248 \(\text{The center of the original circle is }(0,0)\\ \text{The center of the reflected circle is }(8,2) \text{ (can you see why?)}\\ \text{The radius is unchanged. The equation of the new circle is }\\ (x-8)^2 + (y-2)^2 = 25\\ x^2 - 16x + 64 + y^2 - 4y + 4 = 25\\ x^2 +y^2 -16x-4y+43=0\\ a=1,~b=-16,~c=-4,~d=43\)
