Correct Answer: 14

Solution : The common chord of two intersecting circles forms a right angle with the line joining the centres of the two circles.
Let the radii of the two circles as $r_1$ and $r_2$, and the distance between the centres as $d$.
The length of the common chord is given = 24 cm
$l=12\;\mathrm{cm}$
Given that the diameters of the circles are 30 cm and 26 cm.
$r_1=15\;\mathrm{cm}$
$r_1=13\;\mathrm{cm}$
We can use the Pythagorean theorem to find the distance between the centres,
$d = \sqrt{r_1^2 - l^2} + \sqrt{r_2^2 - l^2}$
⇒ $d = \sqrt{r_1^2 - 12^2} + \sqrt{r_2^2 - 12^2}$
⇒ $d = \sqrt{15^2 - 12^2} + \sqrt{13^2 - 12^2}$
⇒ $d = \sqrt{3 \times 27} + \sqrt{1 \times 25}$
⇒ $d=9+5$
⇒ $d=14\;\mathrm{cm}$
Hence, the correct answer is 14.