The equation of circle is x2+y2−6x−10y+P=0... (i) (x−3)2+(y−5)2=(√34−P)2 Centre (3,5) and radius ′r′=√34−P If circle does not touch or intersect the x -axis then radius x<y - coordiante of centre C or √34−P<5 ⇒34−P<25 ⇒P>9 Also if the circle does not touch or intersect x -axis the radius r<x -coordinate of centre C. or √34−P<3⇒34−P<9⇒P>25... (iii) If the point (1,4) is inside the circle, then its distance from centre C<r . or √[(3−1)2+(5−4)2]<√34−P ⇒5<34−K ⇒P<29...(iv) Now all the conditions (ii), (iii) and (iv) are satisfied if 25 <P<29 which is required value of P.