As you are seeking a pair of co-ordinates

you could set up a pair of equations in 2 unknowns.

If the circle centre is the origin, then

where the other co-ordinates are known.

Then the area of the triangle between the origin and the 2 points is

and so, your 2nd equation is, using the area of a triangle for which one vertex is the origin

However, a trigonometric solution will be much simpler.

If you label the acute angle between the negative part of the x-axis, the origin and the red dot "A",

your given angle "B",

and the obtuse angle between the positive part of the x-axis and the blue dot "C",

then use the given co-ordinates to calculate the circle radius and the angle "A",

then calculate "C" and use