Transform $x^{2} + y^{2} - 2 x + 2 y - 3 = 0$ to the parallel axes through $(3, 2)$

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Solution

Let the co-ordinates $(x, y)$ of any point on the given curve change to $(X, Y)$ on shifting the origin to the point $(3, 2)$ , the new axes remaining parallel to the original axes.Then, we get
$x = X + 3$ and $y = Y + 2$
The given equation is $x^{2} + y^{2} - 2 x + 2 y - 3 = 0$
Substituting for $x = X + 3$ and $y = Y + 2$ we get
${(X + 3)}^{2} + {(Y + 2)}^{2} - 2 (X + 3) + 2 (Y + 2) - 3 = 0$
$\Rightarrow X^{2} + 9 + 6 X + Y^{2} + 4 Y + 4 - 2 X - 6 + 2 Y + 4 - 3 = 0$
$\Rightarrow X^{2} + Y^{2} + 4 X + 6 Y + 8 = 0$

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