Distance from curve to origin in R3.?
Find the minimum distance from the origin to the curve 3x^2 + 4xy +3y^2 = 20
It has been so long since I have done this and I am trying to help my nephew you out but I have completely forgot can someone help
- SLv 710 years agoFavorite Answer
this is so weird; I read this problem and putzed with it and could not solve.
However, I woke up this morning with the answer!!
look at both the origin and the form of your equation. Let's say you were blindfolded and someone came along and replaced every x with a y and every y with an x (***including the origin*** - key point) and reshowed you the problem. Ta da!! It's the same problem. so.... your answer has to be of the from x=y (also works if x-->-y and y--->-x)
replace every y and x with X
3X^2 + 4X^2 +3X^2 =20
10 X^2 = 20
distance is sqrt (x^2 +y^2) = sqrt(4)= 2
ta da!!!!!!!! damn I'm good!
let's do a check and see if c works (sqrt(2), sqrt(2)) OR
(-sqrt(2), -sqrt(2) )
3*2 +4*2 +3*2 =20
6+8+6 =20 yep it works
therefore 2 is A distance from (0,0) to the curve (specifically at the points (sqrt(2), sqrt(2)) OR (-sqrt(2), -sqrt(2) )
that ELIMINATES B,D,E as they are all greater than 2.
So the only 2 possible answers are A or C on the basis of "guessing" that ( Sqrt(2), sqrt(2) ) is a point on the curve (it is) that is close to 0,0. But we had a basis for choosing that point as detailed above. It's C on the basis of the symmetry of the equation.
But just to be fanatical with this....I showed that c (2) is a valid distance and since we are looking a minimum distance we can eliminate b,d,e. But just in case you don't believe my explanation - let's just look at a = 1 and eliminate it.
pretend that a is correct (distance is 1); that implies the answer lies on this circle: x^2+y^2 =1
we can write your equation as 3(x^2+y^2) +4xy =20
BUT the stuff in the brackets is 1.
so....3*1 +4xy = 20
xy = 17/4
but I can parameterize my original circle by x= cos(a) and y = sin(a) and rewrite it as cos^2(a) + sin^2(a) = 1 which is a trig identify. In other words, we are looking for a magic angle (a) that does the trick.
so xy =17/4 can be written as
cos(a)*sin(a) = 17/4
there is no a that solves this equation; the reason is the aboslute value of cos(a) and sin(a) by themselves is 1
so cos(a)*sin(a) <1 and cannot reach 17/4
therefore the curve lies further than 1 unit away and answer A is eliminated.Source(s): PhD Biochemist that took a lot of math - specifically including group theory that utilizes symmetry. Your problem is symmetric in x & y.