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Question: Is there a different way to find the conic

Question:Is there a different way to find the conic

Ronan 1267 Maple 2022
geometry conic
February 03 2023
0

This a more a math question than  a maple programming question. But interesting.

I have a point P=[x,y] on a line y= 2+3x. The point maps to Pn=[f(x, y), g(x, y)]. That represents a conic.

I want to find the equation of the conic. I did it by calculating 5 points and that works fine but only because I know I'm looking for a conic. 

Can the problem by solved without having to get the five points?
I copied and pasted the equations for the lines because they were generated by commands in a pagkage I have.

restart

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P := [X, Y]

[X, Y]

``

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Pn := [(5*X+12*Y)*(-384+48*X+55*Y)/(225*X^2+225*Y^2-1800*X-602*Y), (2*(12*X-5*Y))*(-384+48*X+55*Y)/(225*X^2+225*Y^2-1800*X-602*Y)]

[(5*X+12*Y)*(-384+48*X+55*Y)/(225*X^2+225*Y^2-1800*X-602*Y), 2*(12*X-5*Y)*(-384+48*X+55*Y)/(225*X^2+225*Y^2-1800*X-602*Y)]

NULL

``

Y := 2+3*Xn

2+3*X

``

for i to 5 do P || i := eval(Pn, X = i) end do

[181139/51416, -19775/25708]

NULL

l12 := 153621*x*(1/68840)+169429*y*(1/27536)-15067/17210

(153621/68840)*x+(169429/27536)*y-15067/17210

NULL

l34 := -5733*x*(1/473062)+258973*y*(1/946124)+244205/946124

-(5733/473062)*x+(258973/946124)*y+244205/946124

c1 := l12*l34

((153621/68840)*x+(169429/27536)*y-15067/17210)*(-(5733/473062)*x+(258973/946124)*y+244205/946124)

NULL

l13 := 5499*x*(1/2360)+6539*y*(1/944)-793/472

(5499/2360)*x+(6539/944)*y-793/472

l24 := 596115*x*(1/6899489)+7228091*y*(1/6899489)+195364/363131

(596115/6899489)*x+(7228091/6899489)*y+195364/363131

plots:-implicitplot([l12, l34, l13, l24])

c2 := l13*l24

((5499/2360)*x+(6539/944)*y-793/472)*((596115/6899489)*x+(7228091/6899489)*y+195364/363131)

c := simplify(c2*lambda+c1)

(655607277/3256558808)*(x+(2515/846)*y-305/423)*(x+(556007/45855)*y+285532/45855)*lambda-(880709193/32565588080)*(x-(19921/882)*y-18785/882)*(x+(65165/23634)*y-4636/11817)

lambda := solve(eval(c, [x = P5[1], y = P5[2]]), lambda)

-1/4

simplify(c)

-(5039454771/65131176160)*x^2+(1/130262352320)*(-29130506847*y+40315638168)*x-(423368829/3256558808)*y^2+(111619143207/130262352320)*y

plt1 := plot(Y, X = -10 .. 10)

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plt2 := plots:-implicitplot([c])

plots:-display(plt1, plt2, colour = ["Red", "Green"])

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for i in [-7, 13/3, 42*(1/7)] do x := eval(Pn[1], X = i); y := eval(Pn[2], X = i); simplify(c) end do

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Download Q_23-03-04_how_to_find_conic_from_mapping.mw
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