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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

Simran Bhatia , 11 Years ago

Grade 11

1 Answers

Jitender Singh

Last Activity: 10 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

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