SOLUTION: Hello, I have the following general equation for a conic section 0.27067 x^2 + 2.4674 y^2 + 1.63445 xy + 364.02 x -359.4526 y - 25987.84 = 0 This is a puzzle for a geocaching (

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Question 675313: Hello, I have the following general equation for a conic section
0.27067 x^2 + 2.4674 y^2 + 1.63445 xy + 364.02 x -359.4526 y - 25987.84 = 0
This is a puzzle for a geocaching (geocaching.com) puzzle. I need to find the foci, this is rotated and translated. The person who made the puzzle is a mathematics student and it's beyond my abilities. I have tried finding an online solver for foci and directrix of a conic section but I can't seem to get them working.
Thanks in advance
Tim
Answer by jsmallt9(3758)   (Show Source): You can put this solution on YOUR website!
Since you have not already gotten help with this, I'm going to assume that you would appreciate (rather than regret) some partial help. IOW, I'm going to give you some formulas and point you in the right direction but I am now going to solve the whole problem for you.The standard form for conic section equations is (to the best of my knowledge) is:

Putting your equation into this form we get:


Since the "B" is not zero, this conic section (as you already noted) is rotated. To find the angle of rotation, , you use either:
or

Since your A, B and C values will not give us a special angle value for cot or tan, we will need to use our calculator, And since calculators do not (usually) have an inverse tan button we will use the second formula:


Since any angle with this tangent can be used we can simply use the angle we get from the calculator for and not be concerned with any of the other angles whose tan is -0.74403773. You should get:

(Note: No matter how many decimal places we use we will be using decimal approximations. There is no way for us to find a perfectly exact answer (unless our approximations accidentally cancel each other out). So feel free to round these decimals.)

Dividing by 2 we get:


The equations for converting x's and y's to their corresponding values after being rotated by an angle are:

and

where "x" and "y" are the original coordinates and and are the coordinates in the rotated system. With our angle of rotation these equations are:

and


Now it gets messy. Here are the steps to follow:
  1. Use your calculator to find the sin's and cos's above.
  2. Substitute these values into the two equations above.
  3. In the equation replace all three x's with the expression you came up with for "x" in step 2.
  4. In the equation replace all three y's with the expression you came up with for "y" in step 2.
  5. Simplify and transform the equation into the standard form:

    Note: If we were using exact values, the "B" in this new equation should work out to be zero. Because of our approximations it may instead be a number pretty close to zero. If so, just throw out the "B term". (If not, then we've made a mistake somewhere and you will have to backtrack to fix the error.)
  6. Your equation should now have the form:

    With this equation you should be able to...
    1. Identify which type of conic section it is (Hint: It should work out to be an ellipse.)
    2. Transform the equation into the proper form for that type of conic section (the form that tells you the center/vertex, the "a" and "b", etc. (For an ellipse: or
    3. With the proper form you can find the foci and directrix(es).
  7. The foci and directrix you have just computed are expressed in rotated coordinates. If you want answers expressed in terms of the original coordinate system...
    1. For the foci, replace the and in

      and

      with the rotated coordinates you found and simplify.
    2. The rotated directrix(es) should be equation(s) of the form
      = number
      or
      = number
      For the unrotated equations of the directrix(es) we will take the rotated equations and rotate them back . For this use:

      if your rotated directrix was = number. Just replace with "number" from = number.
      or

      if your rotated directrix was = number. Just replace with "number" from = number. IMPORTANT: Notice that the equations above use 18.32531946, not -18.32531946 so you will have to recalculate the sin's and cos's!
As you can tell, this is a lot of work, even with a calculator.
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