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Question

Find the orthogonal trajectories of the family of straight lines- y=mx+1.

A
x2+y2=c
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B
x2+y22y=c
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C
x2+y24y=c
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D
x2+2y22y=c
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Solution

The correct option is B x2+y22y=c
Given equations of the family of straight lines is y=mx+1
Here m is the arbitrary constant or parameter.
Now,differentiating the equation with respect to x gives.,
dydx=m
Substituting the value of m in the equation gives.,
y=x˙y+1
Now,to find the equation of the orthogonal trajectory wwe need to replace dydx by dxdy which gives.,
(y1)dy=xdx
Integrating both sides of the equation gives.,
y22y=x22+c
Rearranging the terms gives.,
x2+y22y=c, where c is a constant.

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