Maxima and minima and finding the radius of the circle

[Matejxx1]

Homework Statement

Find where θ is the biggest (largest) I'll have the picture of the problem included below (pic:1)

Homework Equations

(x-q)²+(y+5/2)²=r²
answer x= 2

The Attempt at a Solution

Hi, so my prefesor gave me this problem and told me to try to solve it. We already did this problem in school and got the answer that x=2.
The trick here is that I am not allowed to use:
the law of cosines

therefore I have tried to circumscribe a circle and found out that the center is located at
S(n,5/2)
and
T₁=(0,4)
T₂=(0,1)
I would now like to know if you guys could help me calculate the perimeter or alternatively if you guys could tell me about some other way to calculate x or θ
thank you

 

[BvU]

One way (depending on your trigonometric and calculus skills) would be to see that $\theta = \theta_1 - \theta_2$ with $\tan\theta_1 = {4\over x}$ and $\tan\theta_2 = {1\over x}$. And, since $0 < \theta<{\pi\over 2}$, $\ \ \ \theta = {\rm max}$ if $\tan\theta = \rm max$

 

[Matejxx1]

Thanks for the answer.
That was the way we did it in the classroom. And the professor also mentioned that this could be solved using the law of cosine. However he asked me if I could find a way to calculate x or θ without using this two ways. I have been trying to do this for about 30 min and I am starting to doubt if this is even possible

 

[blue_leaf77]

You can employ the central angle theorem. But in order to do this, you have to find the correct center of the circle you have constructed on your calculation. The y coordinate can be easily seen to be 5/2, this leaves you the x coordinate of the center. Having found both the coordinate of the circle's center and its radius, you can use the central angle to do the optimization. This method requires a bit more of algebra though.

 

[Matejxx1]

Thanks for the reply. I really appreciate the help.