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# help

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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.

Apr 6, 2019

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$$\text{The center of the original circle is }(0,0)\\ \text{The center of the reflected circle is }(8,2) \text{ (can you see why?)}\\ \text{The radius is unchanged. The equation of the new circle is }\\ (x-8)^2 + (y-2)^2 = 25\\ x^2 - 16x + 64 + y^2 - 4y + 4 = 25\\ x^2 +y^2 -16x-4y+43=0\\ a=1,~b=-16,~c=-4,~d=43$$

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Apr 6, 2019

$$\text{The center of the original circle is }(0,0)\\ \text{The center of the reflected circle is }(8,2) \text{ (can you see why?)}\\ \text{The radius is unchanged. The equation of the new circle is }\\ (x-8)^2 + (y-2)^2 = 25\\ x^2 - 16x + 64 + y^2 - 4y + 4 = 25\\ x^2 +y^2 -16x-4y+43=0\\ a=1,~b=-16,~c=-4,~d=43$$