Question

Find the angle of intersection of the curves y² = x and x² = y.

Answer 1

Solving the given equations

we have y² = x and x² = y

⇒ x⁴ = x or x⁴ – x = 0

⇒ x(x³ – 1) = 0

⇒ x = 0, x = 1

Therefore, y = 0, y = 1

i.e. points of intersection are (0, 0) and (1, 1)

Further y² = x

⇒ `2y "dy"/"dx"` = 1

⇒ `"dy"/"dx" = 1/(2y)`

And x² = y

⇒ `"dy"/'dx"` = 2x.

At (0, 0), the slope of the tangent to the curve y² = x is parallel to y-axis and the tangent to the curve x² = y is parallel to x-axis.

⇒ Angle of intersection = `pi/2`

At (1, 1), slope of the tangent to the curve y² = x is equal to `1/2` and that of x² = y is 2.

tan θ = `|(2 - 1/2)/(1 + 1)| = 3/4`

⇒ θ = `tan^-1 (3/4)`