# Find the angle of intersection of curves x2 + y2 = 4 and (x – 2)2 + y2 = 4.

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

x2 + y2 = 4 and (x – 2)2 + y2 = 4

Put y2 = 4 – (x – 2)2 in x2 + y2 = 4

x2 + 4 – (x – 2)2 = 4

x2 – (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 4th 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 x2 + y2 = 4

12 + y2 = 4

y2 = 3

So here we will get two values of y one in first quadrant and other in 4th 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 m1 and m2 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 x2 + y2 = 4

Differentiating with respect to x  Slope at (1, √3)  Calculating slope for (x – 2)2 + y2 = 4

Differentiating with respect to x   Slope at (1, √3)  Put values of m1 and m2 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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