Equation of the parabola whose axis is y=x, distance from origin to vertex is √ and distance from origin to focus is 2√, is (Focus and vertex lie in 1st quadrant) :
The equation of axis of the parabolain parametric form is x−0cos 45o=y−0sin 45o=√ for A,2√ for S
∴ A is (1, 1) and S is (2, 2) and foot of directrix be z, then A is mid-point of SZ
∴x+22=1,y+22=1,∴z is (0,0).
Equation of directrix is y−0=−1(x−0) or x+y=0
By definition if P(x,y) be any point on the parabola then SP = PM
2[x2+y2−4x−4y+8]=(x+y)2 or x2+y2−2xy=8(x+y−2)