The correct option is A h>2
We have,
2(x−1)2+2(y−1)2=(x+y+2)2
⇒√(x−1)2+(y−1)2=∣∣∣x+y+2√1+1∣∣∣
Clearly, it represents a parabola having its focus at (1,1) and directrix x+y+2=0.
The equation of the axis is y−1=1(x−1) ie, y=x.
Semi latusrectum = Length of perpendicular from (1,1) on the directix.
⇒ Semi latusrectum =∣∣∣1+1+2√1+1∣∣∣=2√2
The coordinates of the vertex are (0,0).
So, the equation of the axis in parametric form is
x−0cosπ4=y−0sinπ4...(i)
We know that three distinct normals can be drawn from a point (h,0) on the axis of the parabola y2=4ax,
if h>2a(=semi latusrectum).
The coordinates of a point on the axis (i) at a distance 2√2 from the vertex are given by
xcosπ4=ysinπ4=2√2⇒x=2,y=2