Tangent to a Parabola with a Given Slope

pbslp.html

Exploration

Show that the line tangent to the parabola with slope m is given by y=m x+p/m.

Approach

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

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 a line parallel to the line and tangent to the parabola. The tangent line has the form expected.

Clear[p];
l2=TangentLines2D[l1,
      p1=Parabola2D[{0,0},p,0]]

Discussion

This is the plot of a numerical example.

Sketch2D[{l2,p1} /. {p->1/2,m->2}]

Graphics saved as "pbslp01.eps".