Proving Orthogonality of Two Families of Parabolas

[azatkgz]

Please,help me with this problem.

Homework Statement

Prove that the two families of parabolas
$$y^2=4a(a-x),a>0$$ and $$y^2=4b(b+x),b>0$$
form an orthogonal net. Specifically, check that for any a, b > 0 these two parabolas
are perpendicular to each other at the points where they intersect.

The Attempt at a Solution

Their tangent spaces at point $$(x_0,y_0)$$ are

$$2y_0(y-y_0)+4a(x-x_0)=0$$

$$2y_0(y-y_0)-4b(x-x_0)=0$$

If they are perpendicular then we have

$$4y_0^2-16ab=0\Rightarrow y_0^2=4ab$$

from the equations of parabolas we have

$$y_0^2=4a(a-x_0)$$

$$y_0^2=4b(b+x_0)$$

if we substitute $$x_0$$

$$y_0^2=4ab$$
So they are perpendicular.

 

[mighty2000]

Dear fellow scientist,

I would be happy to help you with this problem. First, let's define what an orthogonal net is. An orthogonal net is a set of curves that intersect at right angles. In this case, we have two families of parabolas, and we need to prove that they intersect at right angles.

To prove this, we need to show that the tangent lines at the points of intersection are perpendicular. The tangent lines at a point on a parabola can be found by taking the derivative of the equation of the parabola at that point. So, for the first parabola, the tangent line at point (x_0,y_0) is given by:

2y_0(y-y_0)+4a(x-x_0)=0

Similarly, for the second parabola, the tangent line at the same point is given by:

2y_0(y-y_0)-4b(x-x_0)=0

Now, for these two tangent lines to be perpendicular, their slopes must be negative reciprocals of each other. So, let's find the slopes of these two lines by rearranging the equations to the form y=mx+c, where m is the slope. For the first tangent line, we have:

2y_0(y-y_0)+4a(x-x_0)=0

2y_0y-2y_0^2+4ax-4ax_0=0

2y_0y=2y_0^2-4ax+4ax_0

y= y_0-(2a/x_0)x+2a

So, the slope of this line is m_1 = -(2a/x_0).

Similarly, for the second tangent line, we have:

2y_0(y-y_0)-4b(x-x_0)=0

2y_0y-2y_0^2-4bx+4bx_0=0

2y_0y=2y_0^2+4bx-4bx_0

y= y_0-(2b/x_0)x-2b

So, the slope of this line is m_2 = -(2b/x_0).

Now, to prove that these two tangent lines are perpendicular, we need to show that m_1 * m_2 = -1. So, let's substitute the values