Ah....Mathcad......you did very well but you gave up a little too quickly, here........!!!!......notice the following factorization....
15sec^2(x)-32sec(x)+17=0
(15sec(x) - 17) (sec(x) -1) = 0
So either......
sec(x) = 1 which happens at 0 and 360.....but, both of these are out of the requested interval....or....
sec(x) = 17/15 which happens at about 28.072° and about 331.928°
Here's a graph..........https://www.desmos.com/calculator/fpewbunwdj
Notice that the solution of 28.072° is "extraneous"......this will frequently happen when we square both sides of an equation......the only "good" solution in the requested interval occurs at about 331.928°
3x+y=1 and 5x-y=15
Using the first equation, y = 1 - 3x ....and substituting this into the second, we have
5x - (1 - 3x) = 15
5x -1 + 3x = 15
8x - 1 = 15
8x = 16
x = 2 and y = (1 - 3x) = (1 - 3(2)) = (1 - 6) = -5
So the solution point is (2, -5)
And the equation of the circle would be.....
(x - 2)^2 + (y + 5)^2 = 4
Here's a graph.....https://www.desmos.com/calculator/pt2wwhqn4u