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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



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


Grade:11

1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
9 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
askIITians Faculty

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