Tangent to a Hyperbola with Given Slope

hypslp.html

Exploration

Show that the lines tangent to the hyperbola with slope m are given by .

Approach

Construct a line with slope m and use TangentLines2D[ln,quad] to construct the desired tangent lines.

Initialize

To initialize 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

Construct a line with slope m.

Clear[x,y,m];
l1=Line2D[Point2D[x,y],m];

Construct the lines tangent to the hyperbola and parallel to the line.

Clear[a,b];
tln=TangentLines2D[l1,h1=Hyperbola2D[{0,0},a,b,0]]

Show the lines in equation form.

Map[Equation2D[#,{x,y}]&,tln]

Discussion

This is a plot of a numerical example.

Sketch2D[{tln,h1} /. {m->1,a->2,b->1}]

Graphics saved as "hypslp01.eps".