The parabola with directrix x + 2y - 1 = 0 and focus (1, 0) is

To find the equation of the parabola, we use the definition that for any point P(x, y) on the parabola, its distance from the focus S(1, 0) is equal to its distance from the directrix x + 2y - 1 = 0. Step 1: Write the distance from P(x, y) to the focus S(1, 0). The distance PS is given by the distance formula: PS = sqrt((x - 1)^2 + (y - 0)^2) PS = sqrt((x - 1)^2 + y^2) Step 2: Write the distance from P(x, y) to the directrix x + 2y - 1 = 0. The distance PL from a point (x_0, y_0) to a line Ax + By + C = 0 is given by PL = (|Ax_0 + By_0 + C|)/(sqrt(A^2 + B^2)). For the directrix x + 2y - 1 = 0, we have A=1, B=2, C=-1. PL = (|x + 2y - 1|)/(sqrt(1^2 + 2^2)) PL = (|x + 2y - 1|)/(sqrt(1 + 4)) PL = (|x + 2y - 1|)/(sqrt(5)) Step 3: Equate the distances and square both sides. According to the definition of a parabola, PS = PL. sqrt((x - 1)^2 + y^2) = (|x + 2y - 1|)/(sqrt(5)) Squaring both sides to eliminate the square root and absolute value: (x - 1)^2 + y^2 = ((x + 2y - 1)^2)/(5) Step 4: Expand and simplify the equation. Multiply both sides by 5: 5((x - 1)^2 + y^2) = (x + 2y - 1)^2 Expand the left side: 5(x^2 - 2x + 1 + y^2) = 5x^2 - 10x + 5 + 5y^2 Expand the right side using the formula (a+b+c)^2 = a^2+b^2+c^2+2ab+2ac+2bc: (x + 2y - 1)^2 = x^2 + (2y)^2 + (-1)^2 + 2(x)(2y) + 2(x)(-1) + 2(2y)(-1) = x^2 + 4y^2 + 1 + 4xy - 2x - 4y Now, set the expanded left side equal to the expanded right side: 5x^2 - 10x + 5 + 5y^2 = x^2 + 4y^2 + 1 + 4xy - 2x - 4y Step 5: Rearrange the terms to form the general equation of the parabola. Move all terms to one side of the equation: 5x^2 - x^2 + 5y^2 - 4y^2 - 4xy - 10x + 2x + 4y + 5 - 1 = 0 Combine like terms: 4x^2 + y^2 - 4xy - 8x + 4y + 4 = 0 Rearranging the terms in standard form (x^2, xy, y^2, x, y, constant): 4x^2 - 4xy + y^2 - 8x + 4y + 4 = 0 Step 6: Compare the derived equation with the given options. The calculated equation is 4x^2 - 4xy + y^2 - 8x + 4y + 4 = 0. This matches option a). The final answer is a) 4x^2 - 4xy + y^2 - 8x + 4y + 4 = 0. Send me the next one 📸