How do you calculate orthogonal trajectory?
Procedure to find orthogonal trajectory:
- Let f(x,y,c)=0 be the equation of the given family of curves, where c is an arbitrary parameter.
- Differentiate f=0; w.r.t. ‘x’ and eliminate c,ie, form a differential equation.
- Substitute −dydx for dxdy in the above differential equation.
What is orthogonal trajectory of parabola?
Orthogonal trajectory, family of curves that intersect another family of curves at right angles (orthogonal; see figure). Solving this for the orthogonal curve gives the solution y2 + (x2/2) = k, which represents a family of ellipses (shown in red in the figure) orthogonal to the family of parabolas.
What is orthogonal trajectory of circle?
In mathematics an orthogonal trajectory is a curve, which intersects any curve of a given pencil of (planar) curves orthogonally. For example, the orthogonal trajectories of a pencil of concentric circles are the lines through their common center (see diagram).
What are the orthogonal trajectories of the family of curves?
The orthogonal trajectories are the curves that are perpendicular to the family everywhere. In other words, the orthogonal trajectories are another family of curves in which each curve is perpendicular to the curves in original family.
What is the orthogonal trajectories of the given family of straight lines?
By replacing with we see that the orthogonal trajectories for the family of straight lines are concentric circles (Figure ): x 2 + y 2 = R 2 .
What is orthogonal and oblique trajectory?
Since an orthogonal trajectory of the given family intersects each curve of given family atright angels, the slope of the orthogonal trajectory to 7 at x, y! A curve which intersects the curves of the given family at a constant angle a !* )#° is called an oblique trajectory of the given family.
What is the orthogonal trajectories of the family curve y 2 4ax?
The given curve isy2 = 4ax i2ydy/dx = 4aa = ydy/2dx iiEliminating a from Eqs i and ii we get y2 = 4xydy/2dxy = 2xdy/dxReplacing dy/dx by – dx/dy we gety = 2x x -dy/dxydy + 2xdx = 0Integrating we get y2/2 + x2 = cwhich is the required orthogonal trajectory.
How do you find orthogonal trajectory in polar coordinates?
Orthogonal Trajectories in Polar Co-ordinates If φ and φ’ denote the angles which the tangent to the given curve and the trajectory at the point of intersection (r, θ), make with the radius vector to the common point, φ ~ φ’ = (π/2) and so tanφ = – cotφ’. − 1 r d r d θ f o r r d θ d r i .
What are the orthogonal trajectories to the family of curves with equations x2 y2 c2?
The given curve isx2 – y2 = c2 2x – 2ydy/dx = 0dy/dx = x/yReplacing dy/dx by – dx/dy we get-dx/dy = x/ydx/x + dy/y = 0Integrating we get log |x| + log |y| = log c2 xy = c2which is the required orthogonal trajectories.
How do you show two families of curves are orthogonal?
Two curves are said to be orthogonal if their tangent lines are perpendicular at every point of intersection. Two families of curves are said to be orthogonal if every curve in one family is orthogonal to every curve in the other family.
What are oblique trajectories?
Definition. Let F x, y, c!*# be a one parameter family of curves. A curve which intersects the curves of the given family at a constant angle a !* )#° is called an oblique trajectory of the given family.
