Parabola Intersection Angle

pbang.html

Exploration

Show that the parabolas and will cut each other at an angle θ given by

.

Approach

Find the (real) intersection points of the two parabolas (the origin point is an intersection, but the cut angle at the origin is π, so use one of the other (real) angles).  Construct the polars to each parabola at the intersection point (the polars are the tangent lines).  Find the angle between the polars.

Initialize

To inutialize Descarta2D, select the input cell bracket and press SHIFT-Enter.

This initialization assumes that the Descarta2D software has been copied into one of the standard directories for AddOns which are on the Mathematica search path, $Path.

<<Descarta2D`

Solution

Intersect the two parabolas.

Clear[x,y,a,b];
ans=Solve[{y^2==a*x,x^2==b*y},{x,y}]

The first solution is the origin, so it is excluded. The third and fourth solutions are imaginary, so they are ignored. The second solution is the desired one.

p0=Point2D[x,y] /. ans[[2]]

Construct the two parabolas.

{parab1=Loci2D[q1=Quadratic2D[y^2==a*x,{x,y}]],
parab2=Loci2D[q2=Quadratic2D[x^2==b*y,{x,y}]]}

Construct the tangent lines at the points.

{l1=Line2D[p0,q1],l2=Line2D[p0,q2]}

Find the angle between the tangent lines.

eq1=Angle2D[l1,l2]

Discussion

Here's an example with a=1 and b=2.

Sketch2D[{parab1,parab2,l1,l2,p0} /. {a->1,b->2}]

Graphics saved as "pbang01.eps".

The angle is about 36 degrees.

eq1 /. {a->1,b->2}

(eq1 /. {a->1,b->2})/Degree //N