A geometry problem by Israel Sapnu Jr.

Let $P$ be the external point and R be the center of the circle. Let MP=6 be the shortest distance. Let NP=24 be the longest distance.

From the data we get that MR=radius of the circle=9

Now consider the tangent PQ , radius RQ=9 and PR=9+6=15 ,

By applying pythagorus theorem in $\Delta PQR$ , $\angle Q=90°$

$\Longrightarrow PQ=\sqrt { { PR }^{ 2 }-{ QR }^{ 2 } }$

$\Longrightarrow PQ=\sqrt { { 15 }^{ 2 }-{ 9 }^{ 2 } }$

$\Longrightarrow PQ=12\quad units$

Therefore length of tangent = $\boxed{12 units}$

Just to apply the definition power of a circumference