# 3.     The locus of the foot of the perpendicular from the origin to the line which always passes through a fixed point (h,k) is         (a)    parabola (b)    Ellipse (c)    Hyperbola (d)    Circle

Jitender Singh IIT Delhi
8 years ago
Ans:(d) Circle
Let the equation of the line:
$y = mx+c$
Let the foot of perpendicular from (0, 0) to the line be (a, b)
$a = \frac{-mc}{1+m^{2}}$
$b = \frac{c}{1+m^{2}}$
$m = \frac{-a}{b}$
Since line passes through (h, k):
$k = mh +c$
$c = k-mh$
$b = \frac{k-mh}{1+m^{2}}$
$b = \frac{k-(\frac{-a}{b})h}{1+(\frac{-a}{b})^{2}}$
$a^{2}+b^{2}-ah-bk=0$
$a\rightarrow x, b\rightarrow y$
$x^{2}+y^{2}-ax-by=0$
It is a equation of circle.
Thanks & Regards
Jitender Singh
IIT Delhi