Question

show that family of curves r^n=a secn(theta) and r^n=a cosecn(theta) are orthogonal.​

Answer 1

Answer:

Everything is wrong here. Note that if you take a circle, say, r=2, then dr/dθ=0. And yet the slope of the curve at (x,y) is −xy=−cotθ.

Step-by-step explanation:

Answer 2

Two (or more) intersecting curves that are perpendicular at their intersection are said to be orthogonal.

Explanation:

• In mathematics, an orthogonal trajectory is a curve that traverses each curve of a defined pencil of (planar) curves orthogonally.
• Orthogonal trajectories are, for example, lines drawn through the common centre of concentric circles on a pencil.
• Solving differential equations is a good way to figure out how to find orthogonal pathways. The classic method begins with a first-order ordinary differential equation and solves it by separating variables. Both of these tasks appear to be difficult, if not impossible. Numerical procedures must be employed in such cases.
• Orthogonal trajectories are used as curved coordinate systems (i.e. elliptic coordinates) in mathematics, and as electric fields and their accompanying equipotential curves in physics.

• If two curves cut each other orthogonally then at the point of intersection, the product of the tangent of angle b/w radius vector and the curve, of both the curves, will be -1.

• Let the tangent of the angle of the first curve be a and of the second curve be b.

• So,a=2 sin theta /2 cos theta = tan theta

• And b=-cot theta=-1/tan theta

• Therefore, a*b=-1
• Therefore the two curves are orthogonal (QED)