Question

Find the angle of intersection of the curves y = 4 – x² and y = x².

Answer 1

We know that the angle of intersection of two curves is equal to the angle between the tangents drawn to the curves at their point of intersection.

The given curves are y = 4 – x² ....(i) and y = x²  .....(ii)

Differentiating eq. (i) and (ii) with respect to x, we have

`"dy"/"dx"` = – 2x

⇒ m₁ = – 2x

m₁ is the slope of the tangent to the curve (i).

And `"dy"/"dx"` = 2x

⇒ m₂ = 2x

m₂ is the slope of the tangent to the curve (ii).

So, m₁ = – 2x and m₂ = 2x

Now solving equation (i) and (ii) we get

⇒ 4 – x² = x²

⇒ 2x² = 4

⇒ x² = 2

⇒ x = `+- sqrt(2)`

So, m₁ = – 2x

= `-2sqrt(2)` and m₂ = 2x = `2sqrt(2)`

Let θ be the angle of intersection of two curves 

∴ tan θ = `|("m"_2 - "m"_1)/(1 + "m"_1"m"_2)|`

= `|(2sqrt(2) + 2sqrt(2))/(1 - (2sqrt(2))(2sqrt(2)))|`

=  `|(4sqrt(2))/(1 - 8)|`

= `|(4sqrt(2))/(1 - 8)|`

= `|(4sqrt(2))/(-7)|`

= `(4sqrt(2))/7`

∴ θ = `tan^-1 ((4sqrt(2))/7)`

Hence, the required angle is `tan^-1 ((4sqrt(2))/7)`.