20. This problem illustrates one of Descartes’ machines (Fig. 14.21). Here GL is

Question

20. This problem illustrates one of Descartes’ machines (Fig. 14.21). Here GL is a ruler pivoting at G. It is linked at L with a device CNKL that allows L to be moved along AB , always keeping the line K N parallel to itself. The intersection C of the two moving lines GL and KN de- termines a curve. To find the equation of the curve, begin bysettingCB=y,BA=x,andtheconstantsGA=a, KL=b,andNL=c.ThenfindBK,BL,andALinterms of x, y, a, b, and c. Finally, use the similarity relation CB : BL = GA : AL to show that the equation is y2 = cy c xy + ay ac. b Descartes stated, without proof, that this curve is a hyper- bola. Show that he was correct.

Explanation / Answer

We can see that triangle CBK is similar to triangle NLK. Thus, all the sides are proportional. So c/b = y/BK, making BK = yb/c BL=BK-b Therefore, BL = yb/c – b BK = yb/c BL = yb/c - b AK = x + yb/c – b Triangle GAL is similar to triangle CBL Therefore, CB/BL=GA/AL y/(yb/c - b) = a/(x + yb/c - b) y(x + yb/c - b) = a(yb/c - b) yx + y2b/c - yb = yab/c -ab y^2b/c = yab/c - ab + yb - yx y^2 = ya - ac + yc - yxc/b Since the highest degree is two, this curve is a conic section

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