Show two families of curves to be orthogonal.

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Jakotheshadows

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Show the two families of curves to be orthogonal.

ax + by = 0 and x^2 + y^2 = r^2 ... So the family of linear functions which pass through the origin and the family of circles which are centered at the origin. I can mentally imagine why these curves are orthogonal, and I can show this by sketching their graphs, but that is a different part of the problem. I think I am supposed to show algebraically how these curves are orthogonal.
I have found that y' for the family of lines to be y'= -a/b and for the circles y' = -x/y. I am unsure on how to proceed with showing these to be orthogonal in a general sense. Any help is appreciated.
 
for a line of slope -a/b which passes through the origin... (any real constant)
a slope of a line that passes through (0,0) at any point other than the origin is its y (rise) coordinate divided by its x (run) coordinate.. Which doesn't matter at (0,0) because the family of lines doesn't intersect with the circles at x = 0 without (-a/b) being undefined.
ie y = 2x @ x = 4 y = 8... 8 / 4 = 2 the derivative of y. So the derivative of that family of lines is always y/x as long as (-a/b) is defined. Because the derivative of the family of circles centered at the origin is -x/y, and the family of lines described always pass through the origin, slopes are perpendicular at all points of intersection. Thus the two families of curves are orthogonal. I answered my own question. Nothing to see here.
 
Jakotheshadows said:
Show the two families of curves to be orthogonal.

ax + by = 0 and x^2 + y^2 = r^2 ... So the family of linear functions which pass through the origin and the family of circles which are centered at the origin. I can mentally imagine why these curves are orthogonal, and I can show this by sketching their graphs, but that is a different part of the problem. I think I am supposed to show algebraically how these curves are orthogonal.
I have found that y' for the family of lines to be y'= -a/b and for the circles y' = -x/y. I am unsure on how to proceed with showing these to be orthogonal in a general sense. Any help is appreciated.
Click to expand...
What you need to show that at the point of intersection these curves have tangents perpendicular to each other .

point of intersection(x[sub:35udhrsv]1[/sub:35udhrsv],y[sub:35udhrsv]1[/sub:35udhrsv])

y[sub:35udhrsv]1[/sub:35udhrsv] = a/b * (-x[sub:35udhrsv]1[/sub:35udhrsv]) <---- from line 1......................(1)

Slope of the tangent of curve 2 at (x[sub:35udhrsv]1[/sub:35udhrsv],y[sub:35udhrsv]1[/sub:35udhrsv])

2x + 2y * y' = 0

y' = - (x[sub:35udhrsv]1[/sub:35udhrsv])/(y[sub:35udhrsv]1[/sub:35udhrsv])

using (1)

y' = - (x[sub:35udhrsv]1[/sub:35udhrsv])/[a/b * (-x[sub:35udhrsv]1[/sub:35udhrsv])] = b/a

so

m * y'(x[sub:35udhrsv]1[/sub:35udhrsv],y[sub:35udhrsv]1[/sub:35udhrsv]) = -1

Hence the curves are orthogonal

You'll notice pair of ellipse-hyperbola will follow similar relationship.
 
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