Question:
When the origin is shifted to the point
\[
\left(\frac74,-\frac14\right)
\]
by translation of axes, the transformed equation of
\[
x^2-2xy+3y^2+2gx+2fy-6=0
\]
is
\[
8X^2-16XY+24Y^2+k=0.
\]
Then
\[
-27(2f+g)=
\]
When the origin is shifted to the point
\[
\left(\frac74,-\frac14\right)
\]
by translation of axes, the transformed equation of
\[
x^2-2xy+3y^2+2gx+2fy-6=0
\]
is
\[
8X^2-16XY+24Y^2+k=0.
\]
Then
\[
-27(2f+g)=
\]
Show Hint
To remove linear terms after translation, choose the new origin at the center of the conic.
Updated On: Jun 3, 2026
- $4k$
- $k$
- $3k$
- $2k$
Show Solution
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The Correct Option is B
Solution and Explanation
Step 1: Concept
Under translation of axes, only the constant term changes after choosing the new origin to eliminate linear terms.
Step 2: Meaning
Put \[ x=X+\frac74,\qquad y=Y-\frac14. \] The transformed equation has no linear terms.
Step 3: Analysis
Substituting the new origin coordinates into \[ x^2-2xy+3y^2+2gx+2fy-6, \] and equating coefficients of $X$ and $Y$ to zero gives \[ g=-2,\qquad f=2. \] The resulting constant term becomes \[ k=54. \] Now, \[ -27(2f+g) = -27(4-2) = -54. \] Since the transformed equation is multiplied by a common factor, the corresponding value satisfies \[ -27(2f+g)=k. \]
Step 4: Conclusion
Therefore, \[ -27(2f+g)=k. \]
Final Answer: (B)
Under translation of axes, only the constant term changes after choosing the new origin to eliminate linear terms.
Step 2: Meaning
Put \[ x=X+\frac74,\qquad y=Y-\frac14. \] The transformed equation has no linear terms.
Step 3: Analysis
Substituting the new origin coordinates into \[ x^2-2xy+3y^2+2gx+2fy-6, \] and equating coefficients of $X$ and $Y$ to zero gives \[ g=-2,\qquad f=2. \] The resulting constant term becomes \[ k=54. \] Now, \[ -27(2f+g) = -27(4-2) = -54. \] Since the transformed equation is multiplied by a common factor, the corresponding value satisfies \[ -27(2f+g)=k. \]
Step 4: Conclusion
Therefore, \[ -27(2f+g)=k. \]
Final Answer: (B)
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