A Note on Conic Sections and Tangent Circles

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1 Forum Geometricorum Volume FORUM GEOM ISSN A Note on Conic Sections nd Tngent Circles Jn Kristin Huglnd Astrct. This rticle presents result on circles tngent to given conic section nd to ech other. The result is proved using set of prmeteriztions tht cover ll possile scenrios. 1. Introduction The ojective of this rticle is to estlish the following. Theorem 1. Suppose S is conic section of eccentricity 1 if it is n ellipse There exists set S, which is either conic section or union of two conic sections, with the following property. For two circles P nd Q ech tngent to S t two points, nd to ech other externlly t point not on S if S is hyperol, the centers of the two circles R 1 nd R tht re lso tngent externlly to P nd Q nd tngent to S lie on S. For other prolems involving tngent circles, see [1,, 3, ]. The proof of Theorem 1 is sed on explicit prmeteriztions of P, Q nd R 1, given S. It is not necessry to check R seprtely, ecuse of symmetry. In ech cse, the proposed set S is clerly either conic section or union of two conic sections, s climed, ut the following conditions lso need to e verified: The proposed point of tngency etween ny circle mong P, Q nd R 1 nd S lies on S. The proposed center of R 1 lies on the proposed curve i.e., S, or one of its components. c The distnce etween the proposed center of ny circle nd the corresponding proposed point of tngency with S is equl to the proposed rdius. d The line segment from the proposed center of ny circle to the corresponding proposed point of tngency with S is norml to S. e The distnce etween the proposed centers of two mutully tngent circles is equl to the sum of their proposed rdii. Verifying conditions c nd e is generlly done y setting Δx nd to e the differences etween the proposed x- nd y-coordintes respectively for the points in question, nd Δz to e the proposed distnce i.e., the rdius or the sum of the Puliction Dte: Jnury 5, 017. Communicting Editor: Pul Yiu.

2 J. K. Huglnd rdii, nd checking tht Δx + Δz 1 is stisfied. The nottion with Δx nd is lso used when verifying condition d.. Ellipse.1. Prmeteriztion. For the ellipse S: x y + 1,, S is the ellipse x y + 1 where 1 + nd With prmeters α nd u stisfying. sin α, 0 <α π, α u π α, we define three circles P, Q, R 1 with centers nd rdii given elow, nd verify tht their points of tngency with S re s in the rightmost column see Figure 1. P Center Rdius Point of tngency with S cos u α, 0 sin u α cosu α 1, ± Q cos u + α, 0 sin u + α R 1 cos u, 1 cos u sinu cosu+α, ± 1 cos u, 1 cos u cosu α cosu+α Figure 1. Exmple of circles tngent to n ellipse

3 A note on conic sections nd tngent circles 3.. Verifiction. nd re trivil. c For P or Q, wehve Δx + 1 cos u ± α +cos u ± α sin u ± α Δz. For R 1,wehve Δx + cos cos u ± α+ 1 cos u ± α α cos α Δx + 1 cos u cos α + 1 cos u cos α 1 cos u cos α 1 cos u sin u Δz. d The slope of the tngent of S t the point x, y is x, s cn e shown with y sic clculus. For P or Q, tking the point of tngency with positive y-coordinte, we hve 1 cosu±α 1 cosu±α cos u ± α cos u ± α y x. For R 1,wehve Δx cos u 1 cos u 1 cos u cos u y x. e The distnce etween the proposed centers of P nd Q is cos u α cos u + α sin u sin α sin u sin u α+sinu + α, which equls the sum of the proposed rdii.

4 J. K. Huglnd If we insted consider either P or Q together with R 1, the expressions for Δx, nd Δz re s follows: Δx cos u ± α 1 + cos u cos u sin u sin α 1 + cos u cos u cos α +sin u sin α sinucos u sin α + cos u cos α cos u sin u cos u sin α + 1 cos u cos α Δz sin u ± α+ sin u sin u ± cos u sin α+ sin u sin u cos α + cos u sin α ± sin u cos u sin α sin u + 3 sin u + sin u ± sin u cos u sin α., 1 cos u cos, α The reder cn verify tht when we insert these expressions into 1 the terms with ± or cncel out, while collecting the remining terms yields Ξsin u +Ξcos u Ξ, where Ξ , which of course lso implies cncelltion. 3. Prol 3.1. Prmeteriztion. For the prol S: y cx, S is the prol x y c c.

5 A note on conic sections nd tngent circles 5 With prmeter u 1 c, we define the circles P, Q, R 1 in the tle elow, nd verify tht the points of tngency with S re s given in the rightmost column see Figure. P Q R 1 3 Center Rdius Point of tngency with S 0,cu u + 1 c u 1 c ± u u c,cu u 0,cu + u + 1 c u + 1 c ± u + u c,cu + u u 1,cu 1 u c 8c u 1,cu 1 c c Figure. Exmple of circles tngent to prol 3.. Verifiction. Trivil, since ll proposed points of tngency hve the form ± m, cm. With x 3 u 1,wehve c x y c c c u 1 c + 1 8c cu 1 8c s required. c For P or Q, wehve For R 1,wehve Δx + Δx u ± u 1 + u ± 1 Δz c c c. u 1 1 u c + Δz. 8c d The slope of the tngent of S t the point m, cm is c m, nd so the slope of the line segment from the center of the corresponding circle to the point

6 6 J. K. Huglnd of tngency must e 1 c.forp nd Q nd tking the point of tngency with m positive x-coordinte, we hve trivilly Δx 1 c 1 m c m, nd for R 1 we hve lmost eqully trivilly Δx 1 8c 1 1 m c m. e The distnce etween the proposed centers of P nd Q is u, which is equl to the sum of their rdii. If we insted consider P or Q long with R 1,wehve Δx + 9 u 1 16 c + u ± 5 8c. Hyperol 5 16 u ± 5u c + 1 c.1. Prmeteriztion. For the hyperol S: x y 1. 5 u ± 1 Δz. c S is the union of two hyperols S 1 nd S. The first component of S is the hyperol S 1 x y 1, where 1 + +, + +. With prmeter u, we define the circles P, Q, R 1 P Q Center + + R 1 u + + u + u + u + + u +, u +, 0, 0 Rdius + u + u + + u + + u nd verify tht the points of tngency with S re s follows see Figure 3. Point of tngency with S P + u + u, ± + u + u Q + u + + u, ± + u + + u R 1 u +, u + + u

7 A note on conic sections nd tngent circles 7 Figure 3. Exmple of circles tngent to hyperol with center of R 1 on S 1 The second component of S is the hyperol S x y 1, where P Q + +, With prmeter u Center 0, + 0, + R 1 u + +, we define the circles P, Q, R 1 + u u + u + u +, u + Rdius + u + u + u + + u + + nd verify tht the points of tngency with S re s follows see Figure. Point of tngency with S P ± + u u +, + u u Q ± + u + u +, + u + u R 1 u + u, + + u

8 8 J. K. Huglnd Figure. Exmple of circles tngent to hyperol with center of R 1 on S.. Verifiction. For the first component of S, oth P nd Q hve proposed points of tngency with S of the form m, ± m. For the second component, they hve proposed points of tngency of the form ± m +,m. In oth cses, it is cler tht x y 1is stisfied. For R1, we get in oth cses x y In similr fshion, we hve x y + + x y , + 1. c For P nd Q with the first component S 1, strting with the proposed center nd points of tngency with S for P or Q, wehve

9 A note on conic sections nd tngent circles 9 Δx + + u + ± u + + u + ± u + + u + + u ± u + u + + u ± + u + Δz. For R 1,wehve Δx + u u u u + u 1 u + + Δz. + + u + + u + + For P nd Q with the second component S, we hve similrly Δx + + u ± u u ± u + + u + u ± u + u + + u ± + u Δz, nd for R 1 we hve Δx + u u + + u u + u u + + u + + Δz u +

10 10 J. K. Huglnd d The slope of the tngent of S t the point x, y is x, nd so the slope of y the line segment from the center of the corresponding circle to its point of tngency x, y with S must e y.forp or Q for the verifiction for the first component x of S, tking the point of tngency with positive y-coordinte, we hve Δx For R 1,wehve Δx u u + + u + ± u y + u + ± u x. + + u u y x. For the verifiction for the second component of S, tking the point of tngency with positive x-coordinte for P or Q, we hve similrly Δx nd for R 1,wehve Δx u u + + u ± u + u ± u u u y x, y x. e Verifying the condition for P nd Q is rther trivil for oth components of S.ForP or Q long with R 1, with the first component, we hve Δx u + + ± + u u u + ± u + u +,

11 A note on conic sections nd tngent circles 11 Δz u u + u u + + ± u u + u + + u +, + u ± + u u u nd the reder cn verify tht 1 is stisfied. Likewise, for the second component of S,wehve, Δx Δz + + u u + +, u + ± u u + ± u u + u + u + 3 u u + + u + u + ± + u +, u +3 + u + u u + u u ± u + u , for which 1 gin is stisfied.

12 1 J. K. Huglnd 5. Conclusion The prmeteriztions cover ll possile scenrios, nd Theorem 1 is thus proved. Strictly speking, the set of points where R 1 cn hve its center is in generl not the entire conic section, ut suset s implied y the conditions tht we hve included for the prmeters. References [1] H. Fukgw nd D. Pedoe, Jpnese Temple Geometry Prolems, Chrles Bge Reserch Centre, Winnipeg, [] H. Fukgw nd J. F. Rigy, Trditionl Jpnese Mthemtics Prolems of the 18th nd 19th Centuries, SCT Press, Singpore, 00. [3] H. Fukgw nd T. Rothmn, Scred Mthemtics, Jpnese Temple Geometry, Princeton University Press, 008. [] T. Rothmn, Jpnese Temple Geometry, Scientific Americn, 78, My 1998, Jn Kristin Huglnd: Arnstein Arneergs vei 30, Leilighet 308, 1366 Lysker, Norwy E-mil ddress: dmin@neutreeko.net

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