The equation of the auxiliary circle of the e | vedclass

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(A) The given equation of the ellipse is $16 x^{2}+25 y^{2}+32 x-100 y=284$.Rearranging the terms, we get $16(x^{2}+2 x)+25(y^{2}-4 y)=284$.Completing

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$x^{2}+y^{2}+2 x-4 y-20=0$

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(A) The given equation of the ellipse is $16 x^{2}+25 y^{2}+32 x-100 y=284$.Rearranging the terms, we get $16(x^{2}+2 x)+25(y^{2}-4 y)=284$.Completing the square, $16(x^{2}+2 x+1)+25(y^{2}-4 y+4)=284+16+100$.$16(x+1)^{2}+25(y-2)^{2}=400$.Dividing by $400$, we get $\frac{(x+1)^{2}}{25}+\frac{(y-2)^{2}}{16}=1$.Here, $a^{2}=25$. The auxiliary circle of an ellipse $\frac{(x-h)^{2}}{a^{2}}+\frac{(y-k)^{2}}{b^{2}}=1$ is $(x-h)^{2}+(y-k)^{2}=a^{2}$.Thus, the equation of the auxiliary circle is $(x+1)^{2}+(y-2)^{2}=25$.Expanding this, $x^{2}+2 x+1+y^{2}-4 y+4=25$.Therefore, $x^{2}+y^{2}+2 x-4 y-20=0$.

The equation of the auxiliary circle of the ellipse $16 x^{2}+25 y^{2}+32 x-100 y=284$ is

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