Given that a point P is located at (-2.5,4.33) I need to locate the points A and B such that $\frac{PA}{PB} = \frac{4.77}{8}$ and $\angle APB = 55^o $.
A and B must be on the -ve part of the x axis.
I need to get A and B located in order to solve a problem in Control Systems Lag-Lead Compensator design. The problem is found in Modern Control Engineering by Ogata.
HINT: Let $A= (x_a , 0)$ and $B=(x_b, 0)$, with $x_a < x_b < 0$. You need to find $x_a$ and $x_b$. Consider a system of two equations:
In the first you write down the condition regarding the lenght ratio.
The second equation can be written using the Law of cosine, (see https://en.wikipedia.org/wiki/Law_of_cosines), since you know the cosine of the angle $ \angle APB$.