1. Coordinate Systems

Homework

1. Find the equation of the circle of radius \(5\) which passes through the points \(A=(5,-3)\) and \(B=(-2,4)\).

2. Find the equation of the circle with a diameter having the end points: \(A=(3,8)\) and \(B=(-5,2)\)

3. Find the equation of the sphere whose center is \(P=(-4,3,5)\) and is tangent to the \(yz\)-plane.

4. Find the equation of the sphere whose center is \(P=(-1,4,-2)\) and is tangent to the plane \(z=7\).

5. Find the equation of the sphere with a diameter having the end points \(P=(-4,3,5)\) and \(Q=(2,7,-1)\).

6. Plot the polar equation \(r=6\sin\theta\) in a rectangular plot and a polar plot. Then convert to rectangular coordinates and identify the curve.

7. Plot the polar equation \(r=4\sin\theta\cos\theta\) in rectangular and polar forms. Describe the polar plot in words.

8. Convert the polar equation \(r=\csc\theta\cot\theta\) into a rectangular equation and identify the curve.

9. Convert the rectangular equation, \((x^2+y^2)^3=y^4\), into a polar equation and identify the curve.

10. Plot the surface \((r-5)^2+z^2=9\) given in cylindrical coordinates, \((r,\theta,z)\). Notice there is no \(\theta\) in the equation. So the shape is obtained by rotating a curve about the \(z\)-axis. Identify the shape of the surface.

11. Plot the surface, \(\rho=6\cos\phi\), given in spherical coordinates, \((\rho,\phi,\theta)\) for \(0 \lt \phi \lt \pi\). Notice there is no \(\theta\) in the equation. So the shape is obtained by rotating a curve about the \(z\)-axis. Identify the shape of the surface.