In an effort to not cheat on my test, I'm not going to post the actual question, but an example that if you solve could help me with the actual one. Don't give me a final answer, just a jump start I suppose. Thanks!
Find the point on the ellipse x^2 - 2xy + 6y^2 = 10 that is closest to the origin.
2007-11-12 04:35:46 · 4 answers · asked by euchred88 1 in Science & Mathematics ➔ Mathematics
use the equation of the elipse to minimize the distance formula when x1,y1 are set to zero and x2,y2 are the points on the elipse.
2007-11-12 04:44:16 · answer #1 · answered by KEYNARDO 5 · 0⤊ 1⤋
x + y + 2(x² + y²) = ee3d1876b6d67a5db5964e7de39c14b12 (x² + x/2) + (y² + y/2) = 6 (x² + x/2 + -a million±7/ee3d1876b6d67a5db5964e7de39c14b16) + (y² + y/2 + -a million±7/ee3d1876b6d67a5db5964e7de39c14b16) = 6 + -a million±7/8 (x + -a million±7/4)² + (y + -a million±7/4)² = 40 9/8 incorrect, it intersects in a circle, no longer an ellipse. Circle is shifted so the middle is in Q3; hence Qee3d1876b6d67a5db5964e7de39c14b1 could have the min_dist and Q3 could have the max_dist. on condition that x and y are shifted by using the same distance, the line containing the max and min is y = x. Plug in the two and you get 2((x,y) + -a million±7/4)² = 40 9/8 ? (x,y) + -a million±7/4 = ee3d1876b6d67a5db5964e7de39c14b1ee3d1876b6d67a5db5964e7de39c14b1/4 ? (x,y) = [ee3d1876b6d67a5db5964e7de39c14b1ee3d1876b6d67a5db5964e7de39c14b1ee3d1876b6d67a5db5964e7de39c14b1ee3d1876b6d67a5db5964e7de39c14b1]/4 ? (x,y) = ee3d1876b6d67a5db5964e7de39c14b12 or 3/2 max = (ee3d1876b6d67a5db5964e7de39c14b12,ee3d1876b6d67a5db5964e7de39c14b12) min = (3/2, 3/2)
2016-10-16 06:16:27 · answer #2 · answered by ? 4 · 0⤊ 0⤋
I would suggest you eliminate the xy term by rotating the axes through the angle z, where cot 2z = (a-c)/b =(1-6)/-2 = 2.5
Find sin z and cos z and use the rotation formulas to find the equation of the ellipse in the rotated axes (x' -axis and
y'-axis). Then the points closest to the origin will be the endpoints of the minor axis.
2007-11-12 05:08:43 · answer #3 · answered by ironduke8159 7 · 0⤊ 0⤋
Well, Im not sure this is right, but anyway.
First , I decide to try and plot it with a program, and I couldn't
plot it.
Then I tried something else.
I thought, well, there would be a line, y=ax+6, that would intercept it.
SO, i just used value x=1. ANd I came up with 'a' being complex values.
SO, therefore, Im not sure that ellipse exists!
2007-11-12 07:48:41 · answer #4 · answered by yljacktt 5 · 0⤊ 1⤋