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Question

Equation of the parabola whose axis is y=x, distance from origin to vertex is 2 and distance from origin to focus is 22, is (Focus and vertex lie in 1st quadrant) : 


  1. (xy)2=8(x+y2)

  2. (xy)2=4(x+y2)

  3. (x+y)2=4(x+y2)

  4. (x+y)2=2(x+y2)


Solution

The correct option is A

(xy)2=8(x+y2)


The equation of axis of the parabolain parametric form is x0cos 45o=y0sin 45o=2 for A,22 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 y0=1(x0) or x+y=0
By definition if P(x,y) be any point on the parabola then SP = PM
or (x2)2+(y2)2=[x+y2]2
2[x2+y24x4y+8]=(x+y)2 or x2+y22xy=8(x+y2)

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