If length AB, $(x_1,y_1)$ angle POQ are given. Is it possible to get point B $(x_2,y_2)$ ? Yes or No ? Or is the information insufficient?

Question

This is a modification of my previous question here: Is it possible to get the value of angle BAC in this case?

If length AB (say $100$m), $(x_1,y_1)$ say $(-50,0)$ and angle POQ ($30$ degrees) are given. Is it possible to get $(x_2,y_2)$ ? Yes or No ? Or the information still is insufficient to calculate $x_2$ and $y_2$ ?

I can do all sines and cosines myself. But only need some hints.

Answer 1

Let $BC \perp x-axis$ and $BC=x$ then $OC=x\sqrt3$

$x^2+(x\sqrt3 + 50)^2=100^2$

$x^2+25\sqrt3 x-1875=0$

$x=\frac{-25\sqrt3+25\sqrt{15} }{2}$ or $x=26.76$

then

$x_2=x\sqrt3=\frac{-75+75\sqrt{5} }{2}$ or $x_2=46.35$

$y_2=x=\frac{-25\sqrt3+25\sqrt{15} }{2}$ or $y_2=x=26.76$

Answer 2

The information is sufficient.

Draw a circle with centre (x1, y1) and form a locus of its circumference when radius is 100m.

Using point-slope form you can form the equation for the line PO.

Now based on (x1, y1) you can have one or two coordinates possible for B.