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Question
Find the angle of intersection of the curves y2 = x and x2 = y.
Solution
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)`
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