The distance between the centres of two circles of radii 22 cm and 10 cm is 37 cm. If the points of contact of a direct common tangent to these circles are M and Q, then find the length of the line segment MQ.

The correct option is 35 cm.

Step-by-step derivation:

Let the two circles have centers $C_1$ and $C_2$ with radii $r_1$ and $r_2$ respectively.
From the given information:
- Radius of the larger circle, $r_1=22\text{cm}$
- Radius of the smaller circle, $r_2=10\text{cm}$
- Distance between their centers, $d=C_1{C}_2=37\text{cm}$

The line segment $MQ$ represents the direct common tangent to these two circles, where $M$ is the point of contact on the circle of radius $r_1$ and $Q$ is the point of contact on the circle of radius $r_2$.

The formula for the length of a direct common tangent to two circles is given by:
$MQ=\sqrt{d^2-{(r_1-r_2)}^2}$

Substituting the given values into the formula:
$r_1-r_2=22-10=12\text{cm}$

Now, calculate the squares of the values:
$d^2={37}^2=1369$
${(r_1-r_2)}^2={12}^2=144$

Subtract the two squared terms:
$d^2-{(r_1-r_2)}^2=1369-144=1225$

Finally, find the square root of the result:
$MQ=\sqrt{1225}=35\text{cm}$

Thus, the length of the line segment $MQ$ is $35\text{cm}$.