If the point (1, 4) lies inside the circle ${\mathrm{x}}^2+{\mathrm{y}}^2-6\mathrm{x}-10\mathrm{y}+\mathrm{P}=0$ and the circle does not touch or intersect the coordinate axes, then the set of all possible values of P is the interval:

The equation of circle is

$\begin{array}{l}{\mathrm{x}}^2+{\mathrm{y}}^2-6\mathrm{x}-10\mathrm{y}+\mathrm{P}=0\\ (\mathrm{x}-3{)}^2+(\mathrm{y}-5{)}^2=(\sqrt{34-\mathrm{P}}{)}^2\end{array}$    ……(i)

Centre (3, 5) and radius ‘r’ $=\sqrt{34-P}$

If circle does not touch or intersect the x - axis then radius $\mathrm{x}<\mathrm{y}-$ coordinate of centre C or $\sqrt{34-\mathrm{P}}<5$

$\Rightarrow 34-\mathrm{P}<25\Rightarrow \mathrm{P}>9$ …….(ii)

Also if the circle does not touch or intersect x - axis the radius r < x - coordinate of centre C.

or  $\sqrt{34-\mathrm{P}}<3\Rightarrow 34-\mathrm{P}<9\Rightarrow \mathrm{P}>25.......\left(\mathrm{iii}\right)$

If the point (1, 4) is inside the circle, then its distance from centre C < r.

or   $\sqrt{\left[(3-1{)}^2+(5-4{)}^2\right]}<\sqrt{34-\mathrm{P}}\Rightarrow 5<34-\mathrm{K}$

$\Rightarrow \mathrm{P}<29........\left(\mathrm{iv}\right)$

Now all the conditions (ii), (iii) and (iv) are satisfied if 25 < P < 29 which is required value of P.