Orthogonal trajectories Two curves are orthogonal to each othe...

Question

Orthogonal trajectories Two curves are orthogonal to each other if their tangent lines are perpendicular at each point of intersection (recall that two lines are perpendicular to each other if their slopes are negative reciprocals). A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is orthogonal to each curve in the other family. For example, the parabolas y = cx² form orthogonal trajectories with the family of ellipses x²+2y² = k, where c and k are constants (see figure).
Find dy/dx for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories. <IMAGE>

y = cx²; x²+2y² = k, where c and k are constants

Find dy/dx for each equation of the following pairs. Use the derivatives to explain why the families of curves form orthogonal trajectories.

y = cx²; x²+2y² = k, where c and k are constants

Accepted Answer

Welcome back, everyone. For the pair of equations Y equals 10 X2 and X2 + 2 Y2 equals 56, find the first derivative for each. Select the option that correctly explains their relationship. A says they form orthogonal trajectories because the product of their slopes is -1 at the points of intersection. B They are parallel since their slopes are equal. They overlap completely sharing the same path, and D, their slopes add up to 1 indicating that they are supplementary. So first of all, we're going to identify the derivative of Y equals 10x2d. We can factor out the constant. We're differentiating X2, which gives us 10 multiplied by 2 X, and that's 20 X, right? Now, for the second one, we're going to use implicit differentiation. So we're going to take X2 derivative. We're going to add 2 y2 derivative, and we're going to make it equal to the derivative of 56. So for X2, we're going to get 2 X. This is the derivative. We're adding 2 multiplied by 2 Y and Y is a function of X. So according to the chain rule we're multiplying by Y and this is equal to 0 because the derivative of a constant is 0. So we get 4 Y Y is equal to -2 X. We can move 2 X to the right. And now we have to divide both sides by 4 Y to solve 4 Y, right? So Y is equal to -2 X divided by Or Y, which gives us negative x divided by 2y. So we have two expressions and essentially what we can do is multiply those because each represents a slope, right? So we're going to take 20 Xs. This is the expression of the slope of the first function. We have to multiply that by the expression of the slope for the second one, and we get negative X. Divided by 2 Y. Now let's simplify. We're going to get negative 20 x 2 divided by 2y. And this gives us negative. Some x 2 divided by y. Now, the reason why we're multiplying is because if these slopes are orthogonal to each other. Than the product of their slopes. Must be equal to -1. So we want to check if they are orthogonal to each other. Now in this expression, let's recall that Y is equal to 10 x 2, so we can substitute for Y, and this gives us -10 x 2 divided by 10 x 2, and this ratio gives us 1 so we get -1. So indeed the product of their slopes is equal to -1, and we can conclude that the slopes are perpendicular to each other, meaning the correct answer to this problem corresponds to the answer choice A. They form orthogonal trajectories because the product of their slopes is -1 at the points of intersection. Thank you for watching.

Key Concepts

Orthogonal Trajectories

Derivatives and Slopes

Families of Curves

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