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
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
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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.
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.
Updated On: May 20, 2025
- $2x^2 + 3y^2 - z^2 = 25$
- $2x^2 + 3y^2 + z^2 = 25$
- $2x^2 - 3y^2 - z^2 = 25$
- $2x^2 + 3y^2 - z^2 = 50$
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The Correct Option is A
Approach Solution - 1
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:
\[
2X^2 + 3Y^2 - Z^2 - 25 = 0 \Rightarrow 2X^2 + 3Y^2 - Z^2 = 25
\]
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Approach Solution -2
Step 1: Understand the problem
We need to find the transformed equation of the given quadratic surface:
\[ 2x^2 + 3y^2 - z^2 - 8x + 18y + 2z + 9 = 0 \] after translating the coordinate axes to the point \((2, -3, 1)\).
Step 2: Translate the coordinates
Let the new coordinates after translation be:
\[ X = x - 2, \quad Y = y + 3, \quad Z = z - 1 \] So, the original variables become:
\[ x = X + 2, \quad y = Y - 3, \quad z = Z + 1 \]
Step 3: Substitute into the original equation
Substitute \(x = X + 2\), \(y = Y - 3\), and \(z = Z + 1\) into the equation:
\[ 2(X+2)^2 + 3(Y-3)^2 - (Z+1)^2 - 8(X+2) + 18(Y-3) + 2(Z+1) + 9 = 0 \]
Step 4: Expand each term
\[ 2(X^2 + 4X + 4) + 3(Y^2 - 6Y + 9) - (Z^2 + 2Z + 1) - 8X - 16 + 18Y - 54 + 2Z + 2 + 9 = 0 \]
Simplify:
\[ 2X^2 + 8X + 8 + 3Y^2 - 18Y + 27 - Z^2 - 2Z - 1 - 8X - 16 + 18Y - 54 + 2Z + 2 + 9 = 0 \]
Step 5: Combine like terms
- \(8X - 8X = 0\)
- \(-18Y + 18Y = 0\)
- \(-2Z + 2Z = 0\)
- Constants: \(8 + 27 - 1 -16 - 54 + 2 + 9 = (8 + 27) - 1 -16 -54 + 2 + 9 = 35 - 1 - 16 - 54 + 2 + 9 = 34 - 16 - 54 + 2 + 9 = 18 - 54 + 2 + 9 = -36 + 11 = -25\)
So the equation reduces to:
\[ 2X^2 + 3Y^2 - Z^2 - 25 = 0 \] or equivalently,
\[ 2X^2 + 3Y^2 - Z^2 = 25 \]
Step 6: Write the transformed equation
After translation, the equation in the new coordinate system \((X, Y, Z)\) is:
\[ 2X^2 + 3Y^2 - Z^2 = 25 \]
Final answer:
\[ 2x^2 + 3y^2 - z^2 = 25 \] (where \(x, y, z\) represent the new translated axes)
We need to find the transformed equation of the given quadratic surface:
\[ 2x^2 + 3y^2 - z^2 - 8x + 18y + 2z + 9 = 0 \] after translating the coordinate axes to the point \((2, -3, 1)\).
Step 2: Translate the coordinates
Let the new coordinates after translation be:
\[ X = x - 2, \quad Y = y + 3, \quad Z = z - 1 \] So, the original variables become:
\[ x = X + 2, \quad y = Y - 3, \quad z = Z + 1 \]
Step 3: Substitute into the original equation
Substitute \(x = X + 2\), \(y = Y - 3\), and \(z = Z + 1\) into the equation:
\[ 2(X+2)^2 + 3(Y-3)^2 - (Z+1)^2 - 8(X+2) + 18(Y-3) + 2(Z+1) + 9 = 0 \]
Step 4: Expand each term
\[ 2(X^2 + 4X + 4) + 3(Y^2 - 6Y + 9) - (Z^2 + 2Z + 1) - 8X - 16 + 18Y - 54 + 2Z + 2 + 9 = 0 \]
Simplify:
\[ 2X^2 + 8X + 8 + 3Y^2 - 18Y + 27 - Z^2 - 2Z - 1 - 8X - 16 + 18Y - 54 + 2Z + 2 + 9 = 0 \]
Step 5: Combine like terms
- \(8X - 8X = 0\)
- \(-18Y + 18Y = 0\)
- \(-2Z + 2Z = 0\)
- Constants: \(8 + 27 - 1 -16 - 54 + 2 + 9 = (8 + 27) - 1 -16 -54 + 2 + 9 = 35 - 1 - 16 - 54 + 2 + 9 = 34 - 16 - 54 + 2 + 9 = 18 - 54 + 2 + 9 = -36 + 11 = -25\)
So the equation reduces to:
\[ 2X^2 + 3Y^2 - Z^2 - 25 = 0 \] or equivalently,
\[ 2X^2 + 3Y^2 - Z^2 = 25 \]
Step 6: Write the transformed equation
After translation, the equation in the new coordinate system \((X, Y, Z)\) is:
\[ 2X^2 + 3Y^2 - Z^2 = 25 \]
Final answer:
\[ 2x^2 + 3y^2 - z^2 = 25 \] (where \(x, y, z\) represent the new translated axes)
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