How do you find orthogonal trajectory for a family of curves?

Procedure to find orthogonal trajectory

1. Let f(x,y,c)=0 be the equation of the given family of curves, where c is an arbitrary parameter.
2. Differentiate f=0; w.r.t. ‘x’ and eliminate c,ie, form a differential equation.
3. Substitute −dydx for dxdy in the above differential equation.

What is the orthogonal trajectories of family 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 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 find the family of a curve?

The differential equation of the family of curves, x2=4b(y+b),b∈R, is:

1. A. x2=2yy′−x.
2. B. xy′′=y′
3. C. x(y′)2=x−2yy′
4. D. x(y′)2=x+2yy′

What do you know about orthogonal trajectories?

orthogonal trajectory, family of curves that intersect another family of curves at right angles (orthogonal; see figure). In two dimensions, a family of curves is given by the function y = f(x, k), in which the value of k, called the parameter, determines the particular member of the family.

What is the order of differential equation?

The order of a differential equation is defined to be that of the highest order derivative it contains. The degree of a differential equation is defined as the power to which the highest order derivative is raised. The equation (f‴)2 + (f″)4 + f = x is an example of a second-degree, third-order differential equation.

What is the differential equation of the family of parabolas having their vertices at the origin and focus on the Y axis?

The differential equation of family of parabolas with foci at the origin and axis is y=0.

How to find the equation of the family of orthogonal trajectories?

The equation of the family of orthogonal trajectories will be obtained by integrating (iii). Where ‘a’ is a parameter, as the equation of the family of curves. Step 1: Differentiating (i) with respect to x and eliminate the parameter ‘a’ to get the differential equation of the family of curves:

What is the differential equation of the family of orthogonal curves?

The differential equation of the family of orthogonal curves is : − d x d y = y 2 − x 2 2 x y d y d x = − 2 x y y 2 − x 2

When two families are perpendicular to each other at meeting points?

When two families are perpendicular to each other at meeting points, we say that these are mutually orthogonal curves. Furthermore, we say that the family of lines y = Kx is the orthogonal trajectory of the circular curves with the equation x 2 + y 2 = C or vice versa. How can we say that two families are mutually orthogonal to each other?

How do you find the perpendicular slope of an orthogonal family?

If we want to find the orthogonal trajectories, and we know that they’re perpendicular to our family everywhere, then we want a slope for the orthogonal trajectories that is perpendicular to the slope of the original family. To find a perpendicular slope, we take the negative reciprocal (flip it upside down and add a negative sign).

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