Question

If the origin is shifted to the point \(\left(\frac{3}{2}, -2\right)\) by the translation of axes, then the transformed equation of \(2x^2 + 4xy + y^2 + 2x - 2y + 1 = 0\) is:

• A. \(4x^2 + 8xy + 2y^2 - 16 = 0\)
• B. \(2x^2 - 4xy + y^2 = 0\)
• C. \(4x^2 + 8xy + 2y^2 + 9 = 0\)
• D. \(2x^2 - 4xy + y^2 + 16 = 0\)

Hint:

When shifting the origin, use the transformations \(x' = x - x_0\) and \(y' = y - y_0\).
- Substitute the transformed coordinates into the original equation and simplify.

Answer 1

The Correct Option is C

The given equation is: \[ 2x^2 + 4xy + y^2 + 2x - 2y + 1 = 0. \] We are asked to shift the origin to the point \(\left(\frac{3}{2}, -2\right)\). To do this, we use the transformation formulas: \[ x' = x - \frac{3}{2}, \quad y' = y + 2. \] Now, substitute \(x = x' + \frac{3}{2}\) and \(y = y' - 2\) into the equation: \[ 2(x' + \frac{3}{2})^2 + 4(x' + \frac{3}{2})(y' - 2) + (y' - 2)^2 + 2(x' + \frac{3}{2}) - 2(y' - 2) + 1 = 0. \] Expanding all the terms will give: \[ 4x'^2 + 8x'y' + 2y'^2 + 9 = 0. \] Thus, the transformed equation is: \[ \boxed{4x^2 + 8xy + 2y^2 + 9 = 0}. \]