Question : The length of the common chord of two intersecting circles is 24 cm. If the diameters of the circles are 30 cm and 26 cm, then the distance between the centres (in cm) is:
Option 1: 13
Option 2: 14
Option 3: 15
Option 4: 16
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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.
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