Consider the hyperbola
f(x,y) = 3x^2-4y^2-12=0
and the line
g(x,y) = a x + by +c = 0
at tangency points {x_0,y_0} the point normals are aligned
grad f(x_0,y_0)+lambda grad g(x_0,y_0) = vec 0
or
{
(a lambda + 6 x=0), (b lambda - 8 y=0),( c + a x + b y=0)
:}
Solving for x_0,y_0,lambda
x_0 = -(4 a c)/(4 a^2 - 3 b^2), y_0 = -(
3 b c)/( 3 b^2-4 a^2), lambda = (24 c)/(4 a^2 - 3 b^2)
but we are interested in tangent lines which make equal intercepts on the axes. So this implies that a = b then the tangency points are
x_0 = -(4 c)/a, y_0 = (3 c)/a, lambda = (24 c)/a^2
Parameter c is obtained substituting the found values in
f(x_0,y_0)+lambda g(x_0,y_0)=0 giving two solutions
c = pm a
so the tangent lines are
ax + aypm a=0 or
x+y pm 1=0
