3 8.81 x. Score: 0 of 1 pt HW Score: 77.78% (7 of 9 pts) A family of curves form

Question

3 8.81 x. Score: 0 of 1 pt HW Score: 77.78% (7 of 9 pts) A family of curves forms orthogonal trajectories with another family of curves if each curve in one family is perpendicular to each curve in the other family at each point of intersection. Use impicit differentiation if needed to findorach equation of the following pairs. Use he derivatives to explain why the families of curves form orthogonal trajectories. dy dx. a. x2-y2=b2, where a and b are constants b where a and b are constants xy dy Find for xy=ainterms ofx and y only. dx dy dx

Explanation / Answer

1)xy =a

differentiate with respect to x

(1y +x dy/dx)=0

==>m1=dy/dx=-y/x

x2-y2=b2

differentiate with respect to x

2x-2ydy/dx=0

m2=dy/dx =x/y

m1 *m2 =(-y/x)*(x/y)

m1 *m2 =-1

so curves are form orthogonal trajecteries.

y=mx

m1 =dy/dx=m

but y=mx ==>m=y/x

m1 =dy/dx=y/x

x2+y2=a2

differentiate

2x +2y dy/dx =0

==>dy/dx=-2x/2y

==>m2=dy/dx=-x/y

m1*m2=(y/x)*(-x/y)

m1 *m2 =-1

so curves are form orthogonal trajecteries.

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