Question

The number of points on the curve $y=||1-e^x|-2|$ from which mutually perpendicular tangents can be drawn to the parabola $x^2=-4y$ is

Answer 1

Mutually perpendicular tangents on a parabola meet the directrix of the parabola.
So, finding the intersection of the curve and the directrix of the parabola $x^2=-4y$
Directrix $y=1$

Now, $1=||1-e^x|-2|$
$\Rightarrow |1-e^x|=1,3\Rightarrow 1-e^x=\pm 1,\pm 3\Rightarrow 1-e^x=-1,-3(\because e^x>0)\Rightarrow e^x=2,4\therefore x=\ln 2,\ln 4$
Hence, the number of points is $2.$