91Ó°ÊÓ

For the given conic equation, \(x^2 - 4xy + 4y^2 - 2x + 2y = 0\), the coefficients are: - \(A = 1\) - \(B = -4\) - \(C = 4\) - \(D = -2\) - \(E = 2\) - \(F = 0\) Now, let's calculate the discriminant \(\Delta\): $$ \Delta = (-4)^2 - 4(1)(4) = 16 - 16 = 0 $$ Since \(\Delta = 0\), the given conic is a parabola. #tag_title# Step 2: Write the Equation in Standard Form #tag_content# Now, we'll complete the square for the x and y terms. Rearrange the terms: $$ x^2 - 2x - 4xy + 4y^2 + 2y = 0 $$ Group the x and y terms: $$ (x^2 - 2x) - 4xy + (4y^2 + 2y) = 0 $$ Complete the square for the x terms (add and subtract \((\frac{-2}{2})^2 = 1\) ): $$ (x^2 - 2x + 1) - 4xy + 4(y^2 + \frac{1}{2}y) - 1 = 0 $$ Complete the square for the y terms (add and subtract \((\frac{1}{4})^2 = \frac{1}{16}\)" ): $$ (x^2 - 2x + 1) - 4xy + 4(y^2 + \frac{1}{2}y + \frac{1}{16}) - 4(\frac{1}{16}) - 1 = 0 $$ Re-write the equation in the standard form for a parabola: $$ (x - 1)^2 - 4(x - 1)(2y - \frac{1}{2}) = 0 $$ #tag_title# Step 3: Identify Key Properties and Graph the Parabola #tag_content# The standard form of the parabolic equation is: $$ (x - h)^2 = 4p(y - k) $$ By comparing the given equation to the standard form, we can identify these key properties: - Vertex: \((h, k) = (1, \frac{1}{4})\) - Axis of symmetry: A vertical line through the vertex, which is x=1 - Focus: \(p\) is the distance between the vertex and the focus. In our case, \(p = \frac{1}{2}\) and the focus is \((1, \frac{1}{4} + \frac{1}{2}) = (1, \frac{3}{4})\). - Directrix: A horizontal line of length \(p\) distance, and parallel to the axis of symmetry, which is \(y=\frac{1}{4}-\frac{1}{2} = -\frac{1}{4}\). Now, we can graph the parabola using its vertex, axis of symmetry, focus, and directrix.