Unified focus directrix conic equation

I have a problem with this and I'm not sure if the book is having errors so I need to confirm.

The equation R = 42 / ( 6 + 4 cos theta ) can be reduced to
R = 7 / ( 1 + 2/3 cos theta )

Since e = 2/3 and therefore e is less than 1 then the equation resembles that of an ellipse.
The directrix is also vertical and right to the pole of the ellipse.

The vertices of the ellipse is therefore horizontal and cuts at the 0 degree and 180 degree.
Upon substituting 0 and 180 into theta: R = 7 / ( 1 + 2/3 cos theta )

I obtain v ' ( 21, 180degree) and v ( 4.2, 0 degree)

While my answer for v is correct, the book states that the answer to v ' = ( -21, 180 ). Where did the negative came from?

Sorry, I got side-tracked. What I did to double-check was to convert this to a rectangular equation by using the following substitutions:

r**2 = x**2 + y**2; cos(T) = x/r.

So, by multiplying both sides of the equation by 3 + 2cos(T), we get 3r + 2rcos(T) = 21.

Now, subtract 2rcos(T), which is just 2x by my substitution. So, 3r = 21 - 2x. Square both sides

9r**2 = 9x**2 + 9y**2 = 4x**2 - 84x + 441.

Now, if we set y = 0 (i.e. where the vertices on the major axes lie), we can solve x for 4.2 and -21.

Remember what r is with respect to theta. It is r units away in the direction of theta. Theta points left at pi, or 180 degrees, so r will be 21 units to the left of 0 along the y-axis.

In this instance, your book is, in fact, wrong, then. -21 units in the direction of 180 degrees would land you at (21,0).

There is nothing wrong with your answer. Your polar coordinates for the major vertices are correct. Perhaps someone was confusing that point with the rectangular coordinates, which would be (-21, 0).

Have you ever tried to proof-read a mathematics publication? It is easy to miss things like that even if you know your job.