How to convert r=7/(5-5costheta) into rectangular form?

#r=7/(5-5costheta)#

1 Answer
Dean R.
May 7, 2018

That's the sideways parabola #70 x = 25 y^2 - 49 . #

Explanation:

This one is interesting because it just diverges; the minimum of the denominator is zero. It's a conic section; the just diverging I think makes it a parabola. That doesn't matter much, but it does tell us we can get a nice algebraic form without trig functions or square roots.

The best approach is sorta backwards; we use the polar to rectangular substitutions when it seems the other way would be more direct.

#x = r cos theta #

#y = r sin theta #

So # x ^2 + y^2 = r^2(cos^2 theta + sin ^2 theta) = r^2 #

# r = 7/{5 - 5 cos theta}#

We see #r>0.# We start by clearing the fraction.

# 5 r - 5 r cos theta = 7 #

We have an #r cos theta# so that's #x.#

# 5 r - 5 x = 7 #

# 5r = 5 x + 7 #

Our initial observation was #r > 0# so squaring is OK.

#25 r^2 = (5x+7)^2 #

Now we substitute again.

#25 (x^2+ y^2)= (5x+7)^2 #

Technically we've answered the question at this point and we could stop here. But there's still algebra to do, and hopefully a reward at the end: maybe we can show this is actually a parabola.

#25 x^2+ 25 y^2 = 25x^2 + 70 x + 49 #

#25 y^2 - 49 = 70 x #

# x = 1/70 (25 y^2 - 49 ) = 1/70 (5y-7)(5y+7) #

graph{ x = 1/70 ( 25y^2 - 49 ) [-17.35, 50, -30, 30]}

Yes, that's a parabola, rotated #90^circ#from the usual orientation.

Check: Alpha eyball

enter image source here

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