Polar Coordinates — Question 2

Question 2

Problem

Convert $r=6\cos\theta-8\sin\theta$ to Cartesian form and identify the curve.

Question 2 – Solution

Solution

1. Use the polar–Cartesian relationships $$x=r\cos\theta,\qquad y=r\sin\theta,\qquad r^2=x^2+y^2.$$ Equivalent polar coordinates satisfy $$(r,\theta)=(r,\theta+2k\pi)=(-r,\theta+(2k+1)\pi).$$

2. Apply the identity that matches the requested conversion, symmetry test, or intersection, and then check the resulting point or curve in the original polar equation.

3. $r^2=6x-8y$, so $(x-3)^2+(y+4)^2=25$: a circle centered at $(3,-4)$ with radius $5$.