If P is 4cm away from the centre of the circle having a radius of 6cm, it cannot lie on the arc AB of the same circle, because all arcs of circle (by definition) are at a distance of the radius from the centre. That is, in this case, all arcs are 6cm away from the centre. So arc AB, and hence P, must be on a different circle of an unspecified radius.
X hasn't been defined. Is X the centre of the circle having a radius of 6cm?
Is the arc AB on both circles, being the intersection of the two circles? If so, the radius of the P circle has not been specified, so the angle AX̂B (same as APXPB?) cannot be found.
OK. If P is 4cm from the centre (which could be point X). Now let's suppose a chord is drawn through P to intersect the circle at A and B forming the arc AB. The chord is at right angles to the radius. If X is the centre of the circle then AX and BX are radii=6cm. A and B also form the hypotenuses AX and BX of two back-to-back right triangles AXP and BXP, and AXB is an isosceles triangle with perpendicular bisector XP=4cm. The cosine of the angle between AX or BX and XP is 4/6=⅔. The angle AX̂P=BX̂P=48.19°. Therefore the angle AX̂B=96.38°. This is clearly not 60°, so the question needs to be clarified and any missing information provided.
