If the axes are shifted to the point (1, -2) without rotation. What do the equation $2 x^{2} + y^{2} - 4 x + 4 y = 0$ becomes?

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Solution

Let (X, Y) be the coordinates of same point referred to new axes with respect to old coordinates (x,y).

If the origin is shifted at a point (1, -2), then x = X+1 and y = Y-2. Substitute the value of x and y in the equation

$2 x^{2} + y^{2} - 4 x + 4 y = 0$ , we get

2(X+1 $)^{2}$ + (Y-2 $)^{2}$ - 4(X + 1) + 4 (Y-2) = 0

$\Rightarrow 2 (X^{2} + 2 X + 1) + (Y^{2} - 4 Y + 4) - 4 X - 4 + 4 Y - 8 = 0$

$\Rightarrow 2 X^{2} + 4 X + 2 + Y^{2} - 4 Y + 4 - 4 X - 4 + 4 Y - 8 = 0$

$\Rightarrow 2 x^{2} + y^{2} = 6$

$\Rightarrow 2 X^{2} + Y^{2} = 6$

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If the axes are shifted to the point (1, -2) without rotation. What do the equation 2x2+y2−4x+4y=0 becomes?

If the axes are shifted to the point (1, -2) without rotation. What do the equation $2 x^{2} + y^{2} - 4 x + 4 y = 0$ becomes?