Question

By shifting the origin to the point (2, 3) through translation of axes, if the equation of the curve $$ x^2 + 3xy - 2y^2 + 4x - y - 20 = 0 $$ is transformed to the form $$ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, $$ then find $ D + E + F $.

• A. -1
• B. 1
• C. -15
• D. 15

Hint:

When shifting origin, replace \( x \to X + h \) and \( y \to Y + k \), then expand and collect terms to find transformed equation.

Answer 1

The Correct Option is A

Let the new coordinates after translation be: \[ X = x - 2, \quad Y = y - 3 \] Rewrite original equation in terms of \( X, Y \): \[ (x, y) = (X + 2, Y + 3) \] Substitute: \[ (x)^2 = (X + 2)^2, \quad xy = (X + 2)(Y + 3), \quad y^2 = (Y + 3)^2, \quad x = X + 2, \quad y = Y + 3 \] Expand and simplify all terms, combine like terms: After simplification, the new equation will have terms involving \( X, Y \) and constants. The sum of coefficients of linear terms \( D + E \) plus constant term \( F \) equals \(-1\).