Question

If the transformed equation of the equation \[ 2x^2 + 3xy - 2y^2 - 17x + 6y + 8 = 0 \] after translating the coordinate axes to a new origin \((\alpha, \beta)\) is \[ aX^2 + 2h XY + bY^2 + c = 0, \] then find \(3\alpha + c\).

• A. \(h\)
• B. \(2h\)
• C. \(2\beta\)
• D. \(\beta\)

Hint:

Relate the coefficients before and after translation to find constants.

Answer 1

The Correct Option is C

Step 1: Understand translation of origin
Upon translation, the constant term changes, and linear terms vanish. The relation between constants and translation is based on partial derivatives. Step 2: Use formula for translation
For quadratic form \[ Ax^2 + 2Hxy + By^2 + 2Gx + 2Fy + C = 0, \] translating origin to \((\alpha, \beta)\) yields new constant term \[ C' = A\alpha^2 + 2H \alpha \beta + B \beta^2 + 2G \alpha + 2F \beta + C, \] and the new equation \[ aX^2 + 2h XY + b Y^2 + c = 0, \] with \(a = A, h = H, b = B\). Step 3: Given equation coefficients
\[ A = 2, \quad 2H = 3 \Rightarrow H = \frac{3}{2}, \quad B = -2, \quad 2G = -17 \Rightarrow G = -\frac{17}{2}, \quad 2F = 6 \Rightarrow F = 3, \quad C = 8 \] Step 4: Use condition for translation
\[ 3\alpha + c = ? \] From the properties of translation (after eliminating linear terms), it can be shown that: \[ 3\alpha + c = 2 \beta \]