Locus points

Question

Find the locus of points equidistant to the line x + y = 5 and the point (0,0). 

Answer 1

the locus is a parabola slanted 45 degrees with the x axis, opening to the southwest, with vertex = (5/4, 5/4) in quadrant I

the quadratic equation will have an xy term to give it the slant, contrasted with standard vertical or horizontal parabolas

4x^2+2xy+4y^2=25

check the equation by plugging in 3 points on the parabola, the vertex , (0,5/2) and (5/2,0)

check their distances from the focus (0,0) and the directrix

one method is use a standard parabola, y =x^2 tranform it to fit a rotation, convert to polar coordinates, rotate it 135 degrees, then convert it back to rectangular coordinates