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Intersection And Maximum Triangle (Posted on 2008-03-30) Difficulty: 3 of 5
A tangent to the ellipse x2/9 + y2/2 = 1 intersects the circle x2 + y2 = 9 at the points P and Q. It is known that R is a point on the circle x2+ y2 = 9. Each of P, Q and R are located above the x-axis.

For example, the coordinates of R cannot be (3,0), since (3,0) is not located above the x axis.

Determine the maximum area of the triangle PQR.

No Solution Yet Submitted by K Sengupta    
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Some Thoughts Well..... | Comment 1 of 4

The following is wrong. Corrected version to come.

 

Below is a table showing the area, tabulated by the y coordinates of the point of tangency on the ellipse and of point R.

y-coord of
tangent pt.
R        0.100000  0.319036  0.538071  0.757107  0.976142  1.195178  1.414213
0.0000   3.0857986 7.4024063 8.4187058 8.1379706 7.5976373 7.0742322 6.2714960
0.0750   3.0776189 7.3434491 8.3105163 7.9984818 7.4394254 6.9033516 6.0806464
0.1500   3.0684759 7.2822238 8.1998395 7.8567064 7.2791982 6.7306912 5.8881156
0.2250   3.0583667 7.2187241 8.0866685 7.7126378 7.1169499 6.5562460 5.6938986
0.3000   3.0472869 7.1529392 7.9709915 7.5662654 6.9526712 6.3800074 5.4979877
0.3750   3.0352301 7.0848538 7.8527918 7.4175738 6.7863484 6.2019637 5.3003715
0.4500   3.0221877 7.0144484 7.7320478 7.2665430 6.6179641 6.0220993 5.1010354
0.5250   3.0081495 6.9416983 7.6087326 7.1131483 6.4474964 5.8403949 4.8999612
0.6000   2.9931027 6.8665739 7.4828137 6.9573600 6.2749191 5.6568274 4.6971270
0.6750   2.9770327 6.7890404 7.3542529 6.7991428 6.1002011 5.4713693 4.4925069
0.7500   2.9599222 6.7090572 7.2230057 6.6384559 5.9233065 5.2839889 4.2860709
0.8250   2.9417514 6.6265776 7.0890208 6.4752521 5.7441936 5.0946494 4.0777843
0.9000   2.9224975 6.5415481 6.9522397 6.3094776 5.5628151 4.9033090 3.8676075
0.9750   2.9021347 6.4539081 6.8125959 6.1410712 5.3791170 4.7099200 3.6554955
1.0500   2.8806339 6.3635888 6.6700138 5.9699635 5.1930381 4.5144284 3.4413974
1.1250   2.8579621 6.2705124 6.5244083 5.7960763 5.0045095 4.3167733 3.2252555
1.2000   2.8340820 6.1745913 6.3756832 5.6193209 4.8134532 4.1168858 3.0070048
1.2750   2.8089514 6.0757263 6.2237297 5.4395976 4.6197810 3.9146881 2.7865718
1.3500   2.7825229 5.9738054 6.0684250 5.2567935 4.4233936 3.7100924 2.5638736
1.4250   2.7547425 5.8687020 5.9096303 5.0707807 4.2241783 3.5029992 2.3388161
1.5000   2.7255493 5.7602722 5.7471879 4.8814143 4.0220074 3.2932957 2.1112928
1.5750   2.6948737 5.6483524 5.5809181 4.6885293 3.8167354 3.0808534 1.8811824
1.6500   2.6626363 5.5327554 5.4106160 4.4919371 3.6081960 2.8655256 1.6483461
1.7250   2.6287457 5.4132665 5.2360457 4.2914211 3.3961984 2.6471435 1.4126248
1.8000   2.5930965 5.2896373 5.0569352 4.0867308 3.1805220 2.4255125 1.1738342
1.8750   2.5555661 5.1615790 4.8729675 3.8775753 2.9609101 2.2004057 0.9317604
1.9500   2.5160105 5.0287528 4.6837713 3.6636128 2.7370617 1.9715575 0.6861523
2.0250   2.4742595 4.8907577 4.4889068 3.4444391 2.5086204 1.7386530 0.4367124
2.1000   2.4301089 4.7471135 4.2878472 3.2195700 2.2751595 1.5013154 0.1830849
2.1750   2.3833115 4.5972377 4.0799539 2.9884183 2.0361616 1.2590877 0.0751621
2.2500   2.3335624 4.4404126 3.8644399 2.7502605 1.7909889 1.0114066 0.3385605
2.3250   2.2804790 4.2757375 3.6403174 2.5041885 1.5388411 0.7575654 0.6077780
2.4000   2.2235700 4.1020558 3.4063179 2.2490360 1.2786903 0.4966563 0.8836722
2.4750   2.1621866 3.9178399 3.1607664 1.9832631 1.0091794 0.2274806 1.1673752
2.5500   2.0954408 3.7209991 2.9013701 1.7047620 0.7284506 0.0516023 1.4604370
2.6250   2.0220593 3.5085358 2.6248418 1.4105105 0.4338404 0.3429447 1.7650796
2.7000   1.9400967 3.2758698 2.3261586 1.0958910 0.1212791 0.6501409 2.0846984
2.7750   1.8462943 3.0153289 1.9969071 0.7531683 0.2160506 0.9792115 2.4249807
2.8500   1.7343320 2.7120340 1.6207704 0.3673417 0.5913695 1.3418329 2.7969563
2.9250   1.5879918 2.3278020 1.1558760 0.1000846 1.0386052 1.7679685 3.2289299

The maximum area appears when the y-coordinate of the point of tangency is about  just above .5, while the y-coordinate of the point R on the circle is the disallowed zero.

Homing in closer on the point ultimately gives us the following table:

          0.556600  0.556633  0.556667  0.556700  0.556733  0.556767  0.556800
0.0000   8.4226124 8.4226124 8.4226124 8.4226125 8.4226124 8.4226124 8.4226123
0.0003   8.4222448 8.4222448 8.4222449 8.4222448 8.4222448 8.4222447 8.4222447
0.0005   8.4218772 8.4218772 8.4218772 8.4218772 8.4218771 8.4218771 8.4218770
0.0008   8.4215096 8.4215096 8.4215096 8.4215095 8.4215094 8.4215093 8.4215092
0.0010   8.4211420 8.4211419 8.4211419 8.4211418 8.4211417 8.4211416 8.4211415
0.0013   8.4207743 8.4207743 8.4207742 8.4207741 8.4207740 8.4207738 8.4207737
0.0015   8.4204066 8.4204065 8.4204064 8.4204063 8.4204062 8.4204060 8.4204059
0.0018   8.4200389 8.4200388 8.4200387 8.4200386 8.4200384 8.4200382 8.4200380
0.0020   8.4196711 8.4196710 8.4196709 8.4196707 8.4196706 8.4196704 8.4196702
0.0023   8.4193033 8.4193032 8.4193031 8.4193029 8.4193027 8.4193025 8.4193023
0.0025   8.4189355 8.4189354 8.4189352 8.4189350 8.4189348 8.4189346 8.4189343
0.0028   8.4185677 8.4185675 8.4185674 8.4185672 8.4185669 8.4185667 8.4185664
0.0030   8.4181998 8.4181997 8.4181995 8.4181992 8.4181990 8.4181987 8.4181984
0.0033   8.4178320 8.4178318 8.4178315 8.4178313 8.4178310 8.4178307 8.4178304
0.0035   8.4174640 8.4174638 8.4174636 8.4174633 8.4174630 8.4174627 8.4174624
0.0038   8.4170961 8.4170959 8.4170956 8.4170953 8.4170950 8.4170947 8.4170943
0.0040   8.4167281 8.4167279 8.4167276 8.4167273 8.4167269 8.4167266 8.4167262
0.0043   8.4163601 8.4163598 8.4163595 8.4163592 8.4163589 8.4163585 8.4163581
0.0045   8.4159921 8.4159918 8.4159915 8.4159911 8.4159908 8.4159904 8.4159900
0.0048   8.4156240 8.4156237 8.4156234 8.4156230 8.4156226 8.4156222 8.4156218
0.0050   8.4152560 8.4152556 8.4152553 8.4152549 8.4152545 8.4152540 8.4152536
0.0053   8.4148878 8.4148875 8.4148871 8.4148867 8.4148863 8.4148858 8.4148854
0.0055   8.4145197 8.4145193 8.4145189 8.4145185 8.4145181 8.4145176 8.4145171
0.0058   8.4141515 8.4141512 8.4141507 8.4141503 8.4141498 8.4141493 8.4141488
0.0060   8.4137834 8.4137829 8.4137825 8.4137820 8.4137816 8.4137811 8.4137805
0.0063   8.4134151 8.4134147 8.4134142 8.4134138 8.4134133 8.4134127 8.4134122
0.0065   8.4130469 8.4130464 8.4130460 8.4130455 8.4130449 8.4130444 8.4130438
0.0068   8.4126786 8.4126781 8.4126777 8.4126771 8.4126766 8.4126760 8.4126754
0.0070   8.4123103 8.4123098 8.4123093 8.4123088 8.4123082 8.4123076 8.4123070
0.0073   8.4119420 8.4119415 8.4119409 8.4119404 8.4119398 8.4119392 8.4119386
0.0075   8.4115736 8.4115731 8.4115725 8.4115720 8.4115714 8.4115707 8.4115701
0.0078   8.4112052 8.4112047 8.4112041 8.4112035 8.4112029 8.4112023 8.4112016
0.0080   8.4108368 8.4108363 8.4108357 8.4108351 8.4108344 8.4108338 8.4108331
0.0083   8.4104684 8.4104678 8.4104672 8.4104666 8.4104659 8.4104652 8.4104645
0.0085   8.4100999 8.4100993 8.4100987 8.4100980 8.4100974 8.4100967 8.4100959
0.0088   8.4097314 8.4097308 8.4097302 8.4097295 8.4097288 8.4097281 8.4097273
0.0090   8.4093629 8.4093623 8.4093616 8.4093609 8.4093602 8.4093594 8.4093587
0.0093   8.4089944 8.4089937 8.4089930 8.4089923 8.4089916 8.4089908 8.4089900
0.0095   8.4086258 8.4086251 8.4086244 8.4086237 8.4086229 8.4086221 8.4086213
0.0098   8.4082572 8.4082565 8.4082557 8.4082550 8.4082542 8.4082534 8.4082526

Where it seems the maximum area approaches about 8.4226125 when the point of tangency on the ellipse has y-coordinate of about 0.55670 and point R approaches the x-axis on the opposite quadrant (1 vs 2) from the point of tangency.

Peering into the variables of the program, the point of tangency would be at (-2.7578,0.55667) and R would be approaching as close to (3,0) as you dare to get. The intersection points of the tangent line and the circle would be at (-2.9785,0.35846) and (-.037679,2.99976).

So strictly speaking there'd be no maximum, since the interval is open ended (does not include it's endpoint at (3,0)).

Edited on March 30, 2008, 5:07 pm
  Posted by Charlie on 2008-03-30 16:42:55

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