Transform $x^{2} + y^{2} - 3 x + 5 y - 5 = 0$ to the parallel axes through $(1, - 2)$

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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 $(1, - 2)$ , the new axes remaining parallel to the original axes.Then, we get
$x = X + 1$ and $y = Y - 2$
The given equation is $x^{2} + y^{2} - 3 x + 5 y - 5 = 0$
Substituting for $x = X + 1$ and $y = Y - 2$ we get
${(X + 1)}^{2} + {(Y - 2)}^{2} - 3 (X + 1) + 5 (Y - 2) - 5 = 0$
$\Rightarrow X^{2} + 1 + 2 X + Y^{2} - 4 Y + 4 - 3 X - 3 + 5 Y - 10 - 5 = 0$
$\Rightarrow X^{2} + Y^{2} - X + Y - 3 = 0$

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