Find equations of common tangents to two hyperbolas

Question

I want to find equations of common tangents to two hyperbolas $\frac{x^{2}}{5}-\frac{y^{2}}{4}=1$ and $\frac{x^{2}}{4}-\frac{y^{2}}{3}=1$. I think that I should use $y=mx+c$ then I will get something like $$\frac{4 x^{2}-5(m x+c)^{2}}{20}=1$$ for first equation. What should I do for the rest?

Answer 1

The dual conics $X^2/4-Y^2/5-1/20=0, X^2/3-Y^2/4-1/12=0$ do intersect in $(X,Y)=(-1,-1),(-1,1),(1,-1),(1,1),$ making the common tangents $-x-y+1=0,-x+y+1=0,x-y+1=0,x+y+1=0.$