Find the orthogonal trajectory of x2+y2=c
y=mx
Orthogonal trajectory of a family of curves is defined as the family of curves intersecting at right angles. We want to find the orthogonal trajectory of x2+y2=c. This circle has centre as the origin and variable radius. Since the centre is origin for all the circles in this family of curves, any line passing through origin will be a normal to all the circles. We can say that lines of the form y=mx is the orthogonal trajectory of x2+y2=c.
There is a different method for finding orthogonal trajectory of any family of curves. Above method/explanation worked because we know the geometrical interpretation of the curves x2+y2=c.
Let’s look at the second method which would be more helpful in finding the orthogonal trajectory for a general curve
1. Finddydx after differentiating the given equation and eliminate arbitrary constants
2. Replace dydx by −dxdy
3. Solve the differential equation obtained in second step to find the orthogonal trajectory
Let's do the same for the equation given to us.
1. Differentiate x2+y2=c to find dydx and to eliminate c
we get 2x+2ydydx=0
⇒dydx=−xy
2. Replace dydx by −dxdy
⇒−dxdy=−xy
⇒dxdy=xy
3. Solve above differential equation to obtain orthogonal trajectory
dxx=dyy
Integrating both the sides, we get
lnx=lny+k
⇒ln(xy)=k
⇒xy=ek
⇒y=ekx
This equation represents family of straight lines passing through origin as we inferred.
If we replace ek with m, we get y=mx