Line Tangent to a Conic

lntancon.html

Exploration

Use Mathematica to show that the relationship between the coefficients of a line

p x+q y+r=0

tangent to the conic

is given by

(B D-2A E)q r+2(B E-2C D)p r+2(D E-2B F)p q=0.

Approach

Intersect the line and the conic and force the intersection points to be coincident by setting the appropriate terms in the resulting expression to zero.

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

Solve the equation of the line for y.

Clear[p,q,r,x,y];
ans1=Solve[p*x+q*y+r==0,y]

Substitute y into the equation of the conic.

Clear[A1,B1,C1,D1,E1,F1];
eq1=A1*x^2+B1*x*y+C1*y^2+D1*x+E1*y+F1 /. First[ans1]

Solve the equation for x.

ans2=Solve[eq1==0,x]

The line is tangent if the expression under the radical is zero (i.e. the points of tangency are coincident).

expr1=ans2[[1,1,2,3,5,2,1]]

Put the expression into the desired form.

expr2=expr1 //Expand //Factor

Divide both sides by .

expr3=expr2[[2]]

Pick out the coefficients of each of the desired terms (multiplied by -1 to match the desired sign).

Map[-Coefficient[expr3,#]&,
    {p^2,q^2,r^2,q*r,p*r,p*q}]