If the point (1,4) lies inside the circle x2+y2−6x−10y+P=0 and the circle does not touch or intersect the coordinate axes, then the set of all possible values of P is the interval:
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Official solution
Circle: (x−3)2+(y−5)2=34−P, centre (3,5), radius r=34−P.
Not touching/crossing the x-axis: r<5⇒34−P<25⇒P>9.
Not touching/crossing the y-axis: r<3⇒34−P<9⇒P>25.
(1,4) inside: distance to centre <r: (3−1)2+(5−4)2=5<34−P⇒5<34−P⇒P<29.
Combining: 25<P<29.