Answer :

Given: the curves y = 4–x^{2} and y = x^{2}

To find: the angle of intersection of the two curves

Explanation: consider first curve

y = 4–x^{2}

Differentiating the above curve with respect to x we get

Consider second curve

y = x^{2}

Differentiating the above curve with respect to x we get

Given y = x^{2}

Substituting this in other curve equation, we get

x^{2} = 4-x^{2}

⇒ 2x^{2} = 4

⇒ x^{2} = 2

⇒ x = ±√2

When x = √2, we get

y = (√2)^{2}⇒ y = 2

When x = -√2, we get

y = (-√2)^{2}⇒ y = 2

Thus the points of intersection are (√2, 2) and (-√2, 2)

We know angle of intersection can be found by following formula,

i.e.,

Substituting the values from equation (i) and equation (ii), we get

For (√2, 2), the above equation becomes,

Hence the angle of intersection of the curves y = 4–x^{2} and y = x^{2} is

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