In the given, figure, CD and AB are diameters of circle and AB and CD are perpendicular to each other, LQ and SR are perpendiculars to AB and CD respectively. Radius of circle is 5 cm, PB : PA = 2 : 3. What is the length (in cm) of SM?

O is the center of circle and radius is 5 cm.
PB : PA = 2 : 3 (Given)
PB + PA = 10 cm (Diameter)
∴ 2 + 3 = 5 units → 10 cm
              1 unit → 2 cm
∴ PB = 4 cm
PA = 6 cm
OP = AP - OA
AP = 6 cm, OA = 5 cm (Radius)
∴ OP = 6 - 5 = 1 cm
Similarly ON = 1 cm

In ΔONS
Use pythagoras theorem
SO² = NO² + NS²
25 = 1 + NS²
NS² = 24
NS = $$\sqrt {24} $$
∴ SM = NS - MN
∴ MN = OP = 1 cm
SM = $$\sqrt {24} - 1$$
SM = $$\left[ {\left( {2\sqrt 6 } \right) - 1} \right]$$   cm