The Student Room Group

Quick Tangent Question (FP2 Polar)

Okay, suppose you have the curve r=a(1+cosθ)r = a(1+cos\theta) (polar, r,a > 0). If you look in the 2nd and 3rd quadrants you'll see there are two 'bumps', both with a specific point which has a tangent that goes through it perpendicular to the initial line. Now, the angle at each of these points are different, θ=±2π3\theta =\pm \frac{2\pi}{3}. However, the tangents are essentially the same as they both have the same gradient, and they both pass through the same points. My question is, can they be classified as two different tangents, or is it just one?
(edited 12 years ago)
Reply 1
I honestly have no idea, but I will at least try to help as much as I can.

First of all, could you please tell why might this be relevant?

Secondly, I'll give you 2 obvious facts, hopefully helpful though:
- It's just one line (set of points-wise)
- The line is tangent at two points and it's uniquely determined for each. What I mean is that you may refer to it as the tangent at the point theta=2pi/3 and as the tangent at theta=-2pi/3 independently

hope you won't regard this as rubbish :biggrin:
Reply 2
It's needed as proof to see if a curve such as r=(p+qcosθ)r=(p+qcos\theta ) has an 'egg' or 'dimple' shape - which is done by working out how many tangents there are perpendicular to the initial line. Of course, it's for general knowledge as well, just in case I was asked to determine how many tangents there were to a particular curve :smile: .

Quick Reply

Latest