Mutually perpendicular tangents can be drawn from a part on the directrix. So, we should find the number of intersection points of y=||1−ex|−2| and y=1.
Now, y=|1−ex|−2 or y=2−|1−ex|
For y=1
1−ex=±3 or 1−ex=±1
On solving
1−ex=3 (not possible) (∵ex>0)
1−ex=−3 (possible)
1−ex=1 (not possible) (∵ex>0)
1−ex=−1 (possible)
∴ Number of intersections =2.