the angle of intersection of the curves $x^2=4y$ and $y^2=4x$ at point $(0,0)$ is

Question

If two lines are perpendicular, so their slope multiplication is negative. And one of these has slope zero and another is infinity. My question is that the multiplication of infinity to zero can be negative? like sum of no. from 1 to infinity is negative?

Answer 1

The fact that the product of the slopes of two perpendicular lines is $-1$ only applies when both slopes exists. A vertical line has no slope - its slope does not exists. Saying that the slope of a vertical line is infinity and concluding that $\infty \cdot 0 = -1$ requires a severly restricted view of the definition of infinity which does not apply to and does not contradict other situations.

One way around this would be to use tangent vectors instead of slopes. You can parameterize the parabola $x^2=4y$ as $F(t) = (t, \frac 14t^2)$. Then the tangent vector at $F(0)$ is $\vec u = (1,0)$. You can parameterize the parabola $y^2=4x$ as $G(t) = (\frac 14 t^2, t).$ Then the tangent vector at $G(0)$ is $\vec v = (0, 1)$. Since $\vec u \circ \vec v = 0$, it follows that the two tangents are perpendicular.

Answer 2

The first curve is a parabola, tangent to x-axis at origin. The second is also the same parabola rotated around origin through $-90^{\circ} $ and is symmetric with respect to x-axis. Recognizing that the first is an inverse function of the second, there is symmetry of these two curves with respect to line $ x = y. $

So the angle between vertical/horizontal tangents at origin is $90^{\circ} $