How would you find the equations of the asymptotes

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The discussion focuses on finding the equations of asymptotes for hyperbolas, specifically through the equations L_1L_2 + λ = 0 and S + λ = 0, which represent linear functions and straight lines, respectively. There is a query about whether the first two equations are equivalent and how to derive asymptotes from a given curve equation. The third equation, S + 2λ = 0, is identified as representing the conjugate hyperbola if S + λ = 0 denotes the original hyperbola. Clarification is sought on whether the equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = -1\) indeed represents the conjugate hyperbola, confirming that it does. The conversation highlights the complexities involved in understanding these mathematical relationships.
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L_1L_2 +\lambda =0
S+\lambda =0 (Such that D=0)
S+2\lambda =0

In case of a hyperbola, S is the pair of straight lines representing the asymptotes and \lambda is any parameter.

My question is, are the first two equations the same? How would you find the equations of the asymptotes if you were given the equation of the curve.

The third equation is the conjugate hyperbola if S+\lambda =0 represents the original hyperbola. Is there any other way to find the conjugate hyperbola?

If \frac{x^2}{a^2} - \frac{y^2}{b^2} =1 is the equation of the original hyperbola, then does the equation \frac{x^2}{a^2} - \frac{y^2}{b^2} =-1 represent the conjugate hyperbola?
 
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I can make very little sense of this. What are L1 and L2? Are they linear functions? What is S?
 
L_1 and L_2 are linear functions. S represents a pair of straight lines.
 
Yes, to your very last part. I found the rest to be confusing.
 

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