Question

When the origin is shifted to a suitable point, the equation $2x^2+y^2-4x+4y=0$ transforms as $2X^2+Y^2-8X+8Y+18=0$. The point to which origin was shifted is

• A. $(1,2)$
• B. $(1,-2)$
• C. $(-1,2)$
• D. $(-1,-2)$

Answer 1

The correct option is C $(-1,2)$

Let the origin be shifted to $(\alpha ,\beta )$.

Then

$x-\alpha =X$

$y-\beta =Y$

Therefore

$x=X+\alpha$
$y=Y+\beta$

$2x^2+y^2-4x+4y=0$ transforms to

$2(X+\alpha {)}^2+(Y+\beta {)}^2-4(X+\alpha )+4(Y+\beta )=0$

$2X^2+Y^2+2\beta Y+4\alpha X+2{\alpha }^2+{\beta }^2-4X-4\alpha +4Y+4\beta =0$

$2X^2+Y^2+X(4\alpha -4)+Y(2\beta +4)+2{\alpha }^2+{\beta }^2-4\alpha +4\beta =0$

Comparing with,

$2X^2+Y^2-8X+8Y+18=0$

$4\alpha -4=-8$

$4\alpha =-4$
$\alpha =-1$ and

$2\beta +4=8$

$2\beta =4$

$\beta =2$
Thus

$(\alpha ,\beta )=(-1,2)$.