Q. 104.8( 6 Votes )

Find the angle of

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


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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