Let A,B be points on the two half-lines x- 3 |y|= , >0 at a distance of from their point of intersection P . The line segment AB meets the angle bisector of the given half-lines at the point Q . If PQ= 9 2 and R is the radius of the circumcircle of PAB , then ^2 R is equal to ______

Step-by-step solution

The given equations of the half-lines are x - 3 y = for y 0 and x + 3 y = for y 0 . The point of intersection P is obtained by setting y = 0 , which gives x = . Thus, P ( , 0) . The angle bisector of these half-lines is the x-axis ( y = 0 ). The slope of the first half-line is 1 3 , so it makes an angle of 30^ with the positive x-axis. The second half-line has a slope of - 1 3 , making an angle of -30^ with the positive x-axis. Since A and B are at a distance from P , their coordinates can be written using the para