Question

If the angle between the two equal circles
with centres (-2,0),(2,3) is 120° then the
radius of the circle is
1) 5 2)3 3) 1 4) 2​

Answer 1

The radius of the circle is 3 so the correct option is (2).

We have to find the radius of the circle if the angle between two equal circles with centres (-2, 0) and (2, 3) is 120°.

See diagram, Two equal circles of radius r make an angle 120° at C.

here we can apply cosine rule for ∆ABC.

$cosC=\frac{a^2+b^2-c^2}{2ab}$

here AB = c = √{(2 + 2)² + (3)²} = 5

AC = b = r, BC = a = r and ∠C = 120°

∴ cos120° = [r² + r² - 5²]/2(r)(r)

⇒-1/2 = (2r² - 25)/2r²

⇒-2r² = 4r² - 50

⇒6r² = 50

⇒r² = 25/3

⇒r = 5/√3 = 2.8867 ≈ 3

Therefore the radius of the circle is 3. hence option (2) is correct choice.

Answer 2

Step-by-step explanation:

The radius of the circle is 3 so the correct option is (2).

We have to find the radius of the circle if the angle between two equal circles with centres (-2, 0) and (2, 3) is 120°.

See diagram, Two equal circles of radius r make an angle 120° at C.

here we can apply cosine rule for ∆ABC.

����=�2+�2−�22��cosC=2aba2+b2−c2

here AB = c = √{(2 + 2)² + (3)²} = 5

AC = b = r, BC = a = r and ∠C = 120°

∴ cos120° = [r² + r² - 5²]/2(r)(r)

⇒-1/2 = (2r² - 25)/2r²

⇒-2r² = 4r² - 50

⇒6r² = 50

⇒r² = 25/3

⇒r = 5/√3 = 2.8867 ≈ 3

Therefore the radius of the circle is 3. hence option (2) is correct choice.