Question

find the orthogonal trajectories of family of confocal conics x^2/a^2+y^2/b^2+lambda​

Answer 1

Answer:

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Answer 2

Answer: The following equations provide the orthogonal trajectories of the confocal conic family:

x^2/a^2 + y^2/b^2 = -k

where k is a fixed value.

This family of hyperbolas has foci at (c,0), where c2 equals the sum of a2 and b2.

Explanation:

The steps listed below can be used to determine the orthogonal trajectories of the family of confocal conics represented by x2/a2+y2/b2=.

With regard to x, differentiate the above equation to get:

dy/dx = 0 and 2x/a2 + 2y/b2

Calculate (dy/dx):

(b2/a2) (x/y) = - (dy/dx)

Finding the family of curves that fits the following criteria will help us locate the orthogonal trajectory:

Y/X = -1/(dy/dx) = (dy/dx)

which is the same as:

xy = k

where k is a fixed value.

When we enter (dy/dx) from step 2 into the formula xy = k, we obtain:

(b^2/a^2) Solving for x gives us the result:

sqrt(-k(a2/b2)) = x

When we add x to the formula xy = k, we get:

Sqrt(k(b2/a2)) = y

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