To convert the equation 2 x^2+4 x y+5 y^2-4 x-22 y+29=0 to homogeneous form the origin is shifted to the point

Step-by-step solution

The general second-degree equation is given as ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0 . Comparing with 2x^2 + 4xy + 5y^2 - 4x - 22y + 29 = 0 yields: a = 2 , h = 2 , b = 5 , g = -2 , f = -11 , and c = 29 . The center (x₀, y₀) satisfies: 2x₀ + 2y₀ - 2 = 0 and 2x₀ + 5y₀ - 11 = 0 . Solving, x₀ = -2 and y₀ = 3 . Shifting origin to (-2, 3) , the linear terms vanish and the constant becomes: c' = gx₀ + fy₀ + c = (-2)(-2) + (-11)(3) + 29 = 4 - 33 + 29 = 0 . Thus, the homogeneous form is: 2X^2 + 4XY + 5Y^2 = 0 . Final Answe