Question

The angle of intersection of the curves y = x² and x = y² at (0, 0) is __________________.

Answer 1

The given curves are y = x² and x = y².

Let C₁ represents the curve y = x² and C₂ represents the curve x = y².

y = x²

Differentiating both sides with respect to x, we get

$\frac{dy}{dx}=2x$

∴ Slope of the tangent to the curve C₁ at (0, 0) = ${\left(\frac{dy}{dx}\right)}_{\left(0,0\right)}$ = 2 × 0 = 0

So, the the tangent to the curve y = x² at (0, 0) is parallel to the x-axis.

x = y²

Differentiating both sides with respect to x, we get

$1=2y\frac{dy}{dx}$

$\Rightarrow \frac{dy}{dx}=\frac{1}{2y}$

∴ Slope of the tangent to the curve C₂ at (0, 0) = ${\left(\frac{dy}{dx}\right)}_{\left(0,0\right)}$ $=\frac{1}{2\times 0}=\frac{1}{0}=\infty$

So, the tangent to the curve x = y² at (0, 0) is parallel to the y-axis.


Now, the tangent to the curve y = x² at (0, 0) is parallel to the x-axis and tangent to the curve x = y² at (0, 0) is parallel to the y-axis. So, the angle between the tangents to the given curves at (0, 0) is $\frac{\mathrm{\pi }}{2}$.

Thus, the angle of intersection of the given curves = x² and x = y² at (0, 0) is $\frac{\mathrm{\pi }}{2}$.


The angle of intersection of the curves y = x² and x = y² at (0, 0) is $\underline{\frac{\mathrm{\pi }}{2}}$.