We begin with this definition of a hyperbola.(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\left(\overline{F_1 P}-\overline{F_2 P}=2 a\right)\land a>0[/tex]

Perform a few basic algebraic manipulations.

[tex]\sqrt{(c+x)^2+y^2}-\sqrt{(x-c)^2+y^2}=2 a[/tex]

[tex]\sqrt{(c+x)^2+y^2}=2 a+\sqrt{(x-c)^2+y^2}[/tex]

[tex](c+x)^2+y^2=4 a^2+4 a \sqrt{(x-c)^2+y^2}+(x-c)^2+y^2[/tex]

Clearly both sides of this equation must be non-negative. Expand the squares.

[tex]c^2+2 c x+x^2+y^2=4 a^2+4 a \sqrt{(x-c)^2+y^2}+c^2-2 c x+x^2+y^2[/tex]

Again, it appears that both sides must be non-negative. The equation appears to valid without an absolute value symbol.

Now we subtract away some non-negative terms.

[tex]2 c x=4 a^2+4 a \sqrt{(x-c)^2+y^2}-2 c x[/tex]

If [tex]x<0[/tex] there are no real solutions. We can patch it up with absolute value symbols. But when should that restriction first be introduced?

[tex]c x-a^2=a \sqrt{(x-c)^2+y^2}[/tex]

[tex]\mid \frac{c}{a} x-a \mid = \sqrt{(x-c)^2+y^2}[/tex]

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# When to introduce absolute value in hyperbola expression?

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