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Q)

Find the locus of the points of intersection of the mutually perpendicular tangents to a parabola?

$\begin{array}{1 1}(a)\;h+a=0\\(b)\;h+2a=0\\(c)\;h=a\\(d)\;\text{None of these}\end{array}$

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A)
Equation of tangents at point $'t_1'$ and $'t_2'$ are
$t_1y=x+at_1^2$------(1)
$t_2y=x+at_2^2$------(2)
$h=at_1t_2$
$k=a(t_1+t_2)$
Slope of tangents is $\large\frac{1}{t_1}$ & $\large\frac{1}{t_2}$ respectively.
Hence $\large\frac{1}{t_1}\times \large\frac{1}{t_2} $$=-1
$t_1t_2=-1$
Now $h=at_1t_2$
$t_1t_2=-1$
$\Rightarrow h+a=0$
Hence (a) is the correct answer.
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