Find the orthogonal trajectory of the family of curves,y2=x3– 2so, find the value ofyas x→∞.Correct answer is '1'. Can you explain this answer?

Question

Find the orthogonal trajectory of the family of curves,y2=x3&ndash; 2so, find the value ofyas x&rarr;&infin;.Correct answer is '1'. Can you explain this answer?

Answer

To find the orthogonal trajectory of the family of curves y^2 = x^3, we can use the following steps:

1. Differentiate the equation y^2 = x^3 with respect to x to find dy/dx:
2y(dy/dx) = 3x^2

2. Solve for dy/dx:
dy/dx = (3x^2)/(2y)

3. Find the negative reciprocal of dy/dx to obtain the slope of the orthogonal trajectory:
(-1/(dy/dx)) = (-2y)/(3x^2)

4. Set this negative reciprocal slope equal to m and substitute y = mx + c (where c is a constant) into the equation:
(-2y)/(3x^2) = m
y = (-3mx^2)/(2)

5. Rearrange the equation to get the orthogonal trajectory in terms of x and y:
2y = (-3mx^2)
2y + 3mx^2 = 0

Therefore, the orthogonal trajectory of the family of curves y^2 = x^3 is given by the equation 2y + 3mx^2 = 0, where m is a constant.

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