Question:
If the equation \(3x^2 + 4y^2 - xy + k = 0\) is the transformed equation of \(3x^2 + 4y^2 - xy - 5x - 7y + 2 = 0\) after shifting the origin to \((\alpha, \beta)\), then \(\alpha + \beta = k =\)
If the equation \(3x^2 + 4y^2 - xy + k = 0\) is the transformed equation of \(3x^2 + 4y^2 - xy - 5x - 7y + 2 = 0\) after shifting the origin to \((\alpha, \beta)\), then \(\alpha + \beta = k =\)
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When translating axes, plug new coordinates into original equation and simplify constants accordingly.
Updated On: Jun 4, 2025
- \(-2\)
- \(6\)
- \(3\)
- \(-1\)
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The Correct Option is B
Solution and Explanation
When shifting origin to \((\alpha, \beta)\), constant term changes as:
\[
k = 3\alpha^2 + 4\beta^2 - \alpha\beta - 5\alpha - 7\beta + 2
\]
We are given that in transformed equation, only \(k\) remains. Hence all linear terms vanish:
\[
\Rightarrow 6 = \alpha + \beta = k
\]
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