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 \(\alpha + \beta = k =\)

• A. \(-2\)
• B. \(6\)
• C. \(3\)
• D. \(-1\)

Hint:

When translating axes, plug new coordinates into original equation and simplify constants accordingly.

Answer 1

The Correct Option is B

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: \[ \Rightarrow 6 = \alpha + \beta = k \]