Question

Shifting the origin to a suitable point $(X,Y)$ so that the equation $y^2+8x+4y-2=0$ will not contain term in $Y$ and the constant term. Find the co-ordinates of the point to which origin is shifted.

Answer 1

We have,

$y^2=8x+4y-2=0$

If origin is shifted to $(x,y)$ then

$(y+Y{)}^2+8(x+X{)}^2+4(y+Y)-2=0$

$y^2+Y^2+2Yy+8x^2+8X^2+16xX+4y+4Y-2=0$

$2Y+4=0$

$Y=-2$

$Y^2+8X^2+4Y-2=0$

$4+8X^2-8-2=0$

$8X^2=-6$

$X^2=\frac{3}{4}$

$X=\frac{\sqrt{3}}{2}$

$(X,Y)=(\frac{\sqrt{3}}{2},-2)$.

Hence, this is the answer.