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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