Answer :

The given curves are circles

We have to find the angle between circles at point of intersection

Angle between circles at a point means angle between tangents at those point

Let us first find the point of intersection

x^{2} + y^{2} = 4 and (x – 2)^{2} + y^{2} = 4

Put y^{2} = 4 – (x – 2)^{2} in x^{2} + y^{2} = 4

⇒ x^{2} + 4 – (x – 2)^{2} = 4

⇒ x^{2} – (x – 2)^{2} = 0

⇒ (x + x – 2)(x – x + 2) = 0

⇒ 2x – 2 = 0

So here we will get two values of y one in first quadrant and other in 4^{th} but we will consider only the point in first quadrant because the angle will be same at both

⇒ x = 1

Put this x = 1 in x^{2} + y^{2} = 4

⇒ 1^{2} + y^{2} = 4

⇒ y^{2} = 3

So here we will get two values of y one in first quadrant and other in 4^{th} but we will consider only the point in first quadrant because the angle will be same at both

⇒ y = √3

Hence the intersection point is (1, √3)

Now angle between curves or lines is given by where m_{1} and m_{2} are slopes of tangent and θ is required angle between curves

gives us the slope of tangent

Let us find slopes at (1, √3) for both the circles

Calculating slope for x^{2} + y^{2} = 4

Differentiating with respect to x

Slope at (1, √3)

Calculating slope for (x – 2)^{2} + y^{2} = 4

Differentiating with respect to x

Slope at (1, √3)

Put values of m_{1} and m_{2} from (a) and (b) respectively in

⇒ tanθ = |-√3|

⇒ tanθ = √3

⇒ θ = tan^{-1}(√3)

⇒ θ = 60°

Hence angle of intersection is 60°

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