Question : Two circles of radius 13 cm and 15 cm intersect each other at points A and B. If the length of the common chord is 12 cm, then what is the distance between their centres?
Option 1: $\sqrt{145}+\sqrt{184}$
Option 2: $\sqrt{131}+\sqrt{181}$
Option 3: $\sqrt{145}+\sqrt{169}$
Option 4: $\sqrt{133}+\sqrt{189}$
Correct Answer: $\sqrt{133}+\sqrt{189}$
Solution :
Radius is 13 cm and 15 cm
Common chord = 12 cm and Half length of the chord = 6 cm
Let the distance of the centre of the circle with a radius of 15 cm is $x$.
⇒ H$^{2}$ = P$^{2}$ + B$^{2}$
⇒ $ (15)^{2} = (6)^{2} + (B)^{2}$
⇒ 225 = 36 + B$^{2}$
⇒ 225 – 36 = $x^{2}$
⇒ $x$ = $\sqrt{189}$
Let the distance of the centre of the circle with a radius of 13 cm is $y$.
⇒ $H^{2} = P^{2} + B^{2}$
⇒ $(13)^{2} = 6^{2} + y^{2}$
⇒ 169 – 36 = $y^{2}$
⇒ $y$ = $\sqrt{133}$
Now,
⇒ Total distance between the centres = $x +y$ = $\sqrt{189}$ + $\sqrt{133}$
Hence, the correct answer is $\sqrt{189}$ + $\sqrt{133}$.
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