This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Step 1: Define a point (x,y) on the parabola. The definition of a parabola states that any point on the parabola is equidistant from the focus and the directrix. The focus is F=(2,0). The directrix is x=-2. The distance from (x,y) to the focus is d_F = sqrt((x-2)^2 + (y-0)^2). The distance from (x,y) to the directrix x=-2 is d_D = |x - (-2)| = |x+2|. Equating these distances: sqrt((x-2)^2 + y^2) = |x+2| Step 2: Square both sides of the equation to eliminate the square root and the absolute value. (x-2)^2 + y^2 = (x+2)^2 Step 3: Expand the squared terms on both sides. (x^2 - 4x + 4) + y^2 = (x^2 + 4x + 4) Step 4: Simplify and rearrange the equation. x^2 - 4x + 4 + y^2 = x^2 + 4x + 4 Subtract x^2 from both sides: -4x + 4 + y^2 = 4x + 4 Subtract 4 from both sides: -4x + y^2 = 4x Add 4x to both sides: y^2 = 4x + 4x y^2 = 8x This is the equation of the parabola. Step 5: Identify the vertex from the equation. The standard form for a horizontal parabola is (y-k)^2 = 4p(x-h), where (h,k) is the vertex. Comparing y^2 = 8x with (y-k)^2 = 4p(x-h), we can write y^2 = 8x as (y-0)^2 = 8(x-0). Therefore, h=0 and k=0. The equation of the parabola is y^2 = 8x. The vertex of the parabola is (0,0). 3 done, 2 left today. You're making progress.