# Given the two foci and the vertices of an hyperbola and a random line how can one construct the meetings of the curves?

I've made this similar question whose answer doesn't apply to this one.

Given the foci, $$F_1$$ and $$F_2$$, and the vertices, $$V_1$$ and $$V_2$$, of an hyperbola $$\mathcal H$$. How can one construct with straight edge and compass the intersections, $$A$$ and $$B$$, of $$\mathcal H$$ and a random secant line $$\ell$$?

I've tried to figure out one projective transformation of the hyperbola into a circle but I've failed to see it (but on the case in which the hyperbola is rectangular which is the only case I've solved this problem).

• If you solved the problem for a rectangular hyperbola, then you can just transform any hyperbola to a rectangular hyperbola via a dilation with ratio $a/b$, and proceed as in the case of the ellipse. Commented Jun 29, 2021 at 7:54
• ok I finally got it. Commented Jun 29, 2021 at 15:54
• In forgottenbooks.com "Geometrical conics" by Charles Smith 1894, page 11 prop V, and page 26, sections 22 and 23 one can find some nice solutions to the problem. Commented Oct 17, 2023 at 7:28
• @ArneErikson I loved that book Commented Oct 20, 2023 at 19:21

EDIT: one does not need to construct $$f_1$$ neither $$f_2$$.
Draw points $$f_1$$ and $$f_2$$: the foci of the rectangular hyperbola whose vertices are the same of $$\mathcal H$$ aka $$V_1$$ and $$V_2$$. To do so, take the midpoint $$O$$ of $$F_1$$ and $$F_2$$ and draw a square with side $$OV_1$$ and construct a circle centered at $$O$$ and with radius equals to the diagonal of that square. This circle meets line $$F_1F_2$$ in $$f_1$$ and $$f_2$$.
Let $$Q = \ell \cap F_1F_2$$ and construct the line $$\ell '$$ which is the image of $$\ell$$ under the dilation in the $$y$$ (vertical) axis of ratio $$\frac ab$$ (parameters of $$\mathcal H$$). Let $$W$$ be one of the meetings of $$\ell '$$ with the rectangular hyperbola of foci $$f_1$$ and $$f_2$$ and vertices $$V_1$$ and $$V_2$$, then our solution is simply point $$Z$$: the meeting of the perpendicular to $$F_1F_2$$ through $$W$$ with $$\ell$$.
Given line $$s$$ and rectangular hyperbola $$\mathcal R$$ of vertices $$V_1$$ and $$V_2$$ we can construct the meetings of $$s$$ and $$\mathcal R$$ by drawing the circle $$c$$, of diameter $$V_1V_2$$, and the line $$r$$, tangent to $$c$$ in $$V_1$$. Take a random point $$P$$ in $$s$$, let $$T = PV_2 \cap r$$ and let $$P'$$ be such that $$(V_2,T;P,P')=-1$$. Let $$H = r \cap s$$ and $$t = \overleftrightarrow{HP'}$$, then $$\{X_1,X_2\} = t \cap c$$ and so the points $$x_1 = V_2X_1 \cap s$$ and $$x_2 = V_2X_2 \cap s$$ are the meetings of $$s$$ and $$\mathcal R$$