Prove parabolas intersect at right angles (polar eq'ns)

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In summary, the conversation is about proving that two parabolas intersect at right angles. The attempt at a solution involves finding the points of intersection and the slopes of the two equations. The use of a trig identity is suggested to show that the two slopes are perpendicular.
  • #1
subwaybusker
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Homework Statement


Show that the parabolas r=c/(1+cosθ)and r'=d/(1-cosθ) intersect at right angles.

The Attempt at a Solution



I found the points of intersection by setting the two equations equal, to which I got:
cosθ = (c- d)/(c+d)
θ = cos^-1[(c- d)/(c+d)]

then i tried to find the slope of the two equations:

x=dcosθ/1-cosθ ; y=dsinθ/1-cosθ

dy/dθ = [dcosθ(1-cosθ)-(sinθ)dsinθ] / (1-cosθ)^2 = d(cosθ-1)/(1-cosθ)^2
dx/dθ= [-dsinθ(1-cosθ)-(sinθ)dcosθ] / (1-cosθ)^2 = -dsinθ/(1-cosθ)^2

dy/dx=cosθ-1/-sinθ

x=ccosθ/1+cosθ ; y=csinθ/1+cosθ

dy'/dθ = [-csinθ(1+cosθ)-(-sinθ)ccosθ] / (1+cosθ)^2 = -csinθ/(1+cosθ)^2
dx'/dθ= [ccosθ(1+cosθ)-(-sinθ)csinθ] / (1+cosθ)^2 = c(cosθ+1)/(1+cosθ)^2

dy/dx=cosθ+1/-sinθ

Then I don't know what to do
 
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  • #2
Two slopes are perpendicular if their product is -1. Are they? Use a trig identity.
 
  • #3
thanks!
 
  • #4
You don't have to assume that. sin(x)^2+cos(x)^2=1. So cos(x)^2-1=-sin(x)^2. ALWAYS. It's an identity.
 

Related to Prove parabolas intersect at right angles (polar eq'ns)

What is a parabola?

A parabola is a symmetrical curve formed by the intersection of a cone and a plane parallel to its side. It is defined as the set of points that are equidistant from a fixed point (focus) and a fixed line (directrix).

What is a polar equation?

A polar equation is a mathematical representation of a curve or shape using polar coordinates instead of the typical Cartesian coordinates (x and y). In polar coordinates, a point is represented by its distance from the origin and the angle it makes with the positive x-axis.

How do you prove that two parabolas intersect at right angles using polar equations?

To prove that two parabolas intersect at right angles, we can use the fact that the angle of intersection between two curves is equal to the difference in their slopes at that point. We can find the slopes of the two parabolas by taking the derivative of their polar equations and setting them equal to each other. If the resulting equation has a solution of 90 degrees, then we can conclude that the two parabolas intersect at right angles.

Is there a specific condition that must be met for two parabolas to intersect at right angles?

Yes, for two parabolas to intersect at right angles, their axes must be perpendicular to each other. This means that the directrices of the two parabolas must be parallel to each other, and the distance between their foci must be equal to the distance between their directrices.

Can this proof be applied to other types of equations?

Yes, this proof can be applied to any two curves or shapes represented by polar equations. As long as we can find the slopes of the two curves and show that their difference is equal to 90 degrees, we can prove that they intersect at right angles.

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