Finding the Foci of a Quadratic: Working with y^2=1-2x^2

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To find the foci of the quadratic equation y^2 = 1 - 2x^2, it needs to be rewritten in the standard ellipse form x^2/a^2 + y^2/b^2 = 1. The equation can be rearranged to 2x^2 + y^2 = 1, which must match the standard form. By comparing coefficients, it is determined that a^2 = 1/2 and b^2 = 1, leading to the values of a and b necessary for the ellipse. The positions of the foci can then be derived from these values, as detailed in standard textbooks on the subject. Understanding the relationship between a and b is crucial for locating the foci accurately.
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y^2= 1- 2x^2

im supposed to put it in x^2/a^2 + y^2/b^2 = 1 to find the foci and so forth.

when i try to set it up i get 2x^2 + y^2 = 1. I am not sure where to go from here. if i multiply or divide to set the problem up properly the 1 will change and my equation will be messed up. can someone show me what my next step is?
 
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Try match the equation for your particular ellipse, 2x2 + y2 = 1, with the general equation for an ellipse which you have also written up. What value must a and b have in order to match your equation?
 
should I put everything over 1? so that the 1 stays the same?
 
I think you misunderstand. If you compare the equation 2x2 + y2 = 1 with x2/a2 + y2/b2 = 1, then what value must a and b have for the two equations to have the same set of solutions, that is, for them to express the same ellipse?

Hint: 2x2 must be equal to x2/a2 for all x, and likewise y2 must equal y2/b2 for all y. Solving these two equations will to get you a and b, and from these you should be able to derive the position of the loci (you may want to look in your textbook on how this position is related to the value of a and b).
 

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