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Question Number 165122 by mr W last updated on 26/Jan/22

Commented by mr W last updated on 26/Jan/22

three vertices of the square lie on  the parabola. find the locus of the  fourth vertex inside the parabola.

threeverticesofthesquarelieontheparabola.findthelocusofthefourthvertexinsidetheparabola.

Commented by aleks041103 last updated on 26/Jan/22

does the square have a defined side length

doesthesquarehaveadefinedsidelength

Commented by mr W last updated on 26/Jan/22

no, it mustn′t have fixed side length.

no,itmustnthavefixedsidelength.

Answered by aleks041103 last updated on 26/Jan/22

The curve is the central branch of  8+92X^2 −18X^4 +X^6 −28Y+116X^2 Y−11X^4 Y+30Y^2 +19X^2 Y^2 −9Y^3 =0  where X=(x/a) and Y=(y/a)

Thecurveisthecentralbranchof8+92X218X4+X628Y+116X2Y11X4Y+30Y2+19X2Y29Y3=0whereX=xaandY=ya

Commented by aleks041103 last updated on 26/Jan/22

Commented by aleks041103 last updated on 26/Jan/22

No matter what I do, I can′t separate  the three branches...

NomatterwhatIdo,Icantseparatethethreebranches...

Commented by aleks041103 last updated on 26/Jan/22

Commented by mr W last updated on 27/Jan/22

great work sir!  i think the complete locus has indeed  three branches, since the fourth  vertex can be inside of the parabola  as well as outside of the parabola.

greatworksir!ithinkthecompletelocushasindeedthreebranches,sincethefourthvertexcanbeinsideoftheparabolaaswellasoutsideoftheparabola.

Commented by mr W last updated on 27/Jan/22

Commented by mr W last updated on 27/Jan/22

for both cases there is a minimum  square (blue ones).

forbothcasesthereisaminimumsquare(blueones).

Commented by mr W last updated on 27/Jan/22

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