Question

The transformed equation of $2x^2-3y^2+z^2+4x+6y-4z-2=0$ when the axes are translated to the point $(-1,1,2)$ is

Answer 1

The equation given is

${2x}^2-{3y}^2+z^2+4x+6y-4z-2=0⟶\left(1\right)$

Now the origin is translated to the point $(-1,1,2)$

The new coordinates in the transformed coordinates system are associated with the old one as

$x^1=x-x_0=x+1\Rightarrow x=(x^1-1)$

$y^1=y-y_0=y-1\Rightarrow y=(y^1+1)⟶\left(2\right)$

$z^1=z-z_0=z-2\Rightarrow z=(z^1+2)$

Substituting in equation $\left(1\right)$ we get ;

$2{(x^1-1)}^2-3{(y^1+1)}^2+{(z^1+2)}^2+4(x^1-1)+6(y^1+1)-4(z^1+2)-2=0$

$\Rightarrow 2(x^{{}^{\prime }2}+1-{2x}^1)-3(y^{{}^{\prime }2}+1+2y)+(z^{{}^{\prime }2}+4+4z^1)+{4x}^1-4+{6y}^1+6-4z^1-8-2=0$

$\Rightarrow {2x}^{{}^{\prime }2}-3y^{{}^{\prime }2}+z^{{}^{\prime }2}-4x^1+{4x}^1-{6y}^1+{6y}^1+{4z}^1-{4z}^1+2-3+4-4+6-8-2=0$

$\Rightarrow {2x}^{{}^{\prime }2}-3y^{{}^{\prime }2}+z^{{}^{\prime }2}-5=0$

$\therefore$ the transformed equation is

${2x}^{{}^{\prime }2}-{3y}^{{}^{\prime }2}+z^{{}^{\prime }2}-5=0$