Question

The transformed equation of $2x^2 + 3y^2 - z^2 - 8x + 18y + 2z + 9 = 0$ when the axes are translated to the point $(2,-3,1)$ is

• A. $2x^2 + 3y^2 - z^2 = 25$
• B. $2x^2 + 3y^2 + z^2 = 25$
• C. $2x^2 - 3y^2 - z^2 = 25$
• D. $2x^2 + 3y^2 - z^2 = 50$

Hint:

Axis Translation.
Translate using $x = X + h$, $y = Y + k$, $z = Z + l$. After substitution, expand and simplify. Linear terms vanish if the origin is moved to the center.

Answer 1

The Correct Option is A

Use: $x = X + 2$, $y = Y - 3$, $z = Z + 1$. Substitute into the given equation: \[ 2(X+2)^2 + 3(Y-3)^2 - (Z+1)^2 - 8(X+2) + 18(Y-3) + 2(Z+1) + 9 = 0 \] Simplify all terms to get: \[ 2X^2 + 3Y^2 - Z^2 - 25 = 0 \Rightarrow 2X^2 + 3Y^2 - Z^2 = 25 \]