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

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Solution

Mutually perpendicular tangents can be drawn from a part on the directrix. So, we should find the number of intersection points of $y = | | 1 - e^{x} | - 2 |$ and $y = 1$ .
Now, $y = | 1 - e^{x} | - 2$ or $y = 2 - | 1 - e^{x} |$
For $y = 1$
$1 - e^{x} = \pm 3$ or $1 - e^{x} = \pm 1$
On solving
$1 - e^{x} = 3$ (not possible) $(∵ e^{x} > 0)$
$1 - e^{x} = - 3$ (possible)
$1 - e^{x} = 1$ (not possible) $(∵ e^{x} > 0)$
$1 - e^{x} = - 1$ (possible)
$∴$ Number of intersections $= 2$ .

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