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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.

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    $\begingroup$ Isn't there any other condition or information? The problem as written has infinity solutions (A,B) $\endgroup$ – user376343 Dec 2 '18 at 9:45
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    $\begingroup$ That information alone is not enough. You can rotate the triangle around point $P$ however you like and get infinitely many different values for $A$ and $B$. Or are you looking for a general expression? $\endgroup$ – platty Dec 2 '18 at 9:46
  • $\begingroup$ Sorry my bad you both are right I forgot to mention that A and B must be on the x-axis (-ve part of x-axis to be specific) $\endgroup$ – Hasan Hammoud Dec 2 '18 at 9:48
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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$.

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  • $\begingroup$ Thanks for your answer thats exactly what I did however the equations got very messy. I was thinking if there is a more mathematically easier way of getting this solved $\endgroup$ – Hasan Hammoud Dec 2 '18 at 10:21

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