3. Hyperbolas (Conic Sections)

L3CALC

Author: Juyoung Park & Sedecember juyoung.parkk8@gmail.com & a3060100s@gmail.com

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A definition of a hyperbola is the set of all points in a plane where the difference of the distances to two fixed points (called the foci) is always constant.

It looks pretty similar to the definition of an ellipse, and we can anticipate that the final expression will be pretty similar to the ellipse expression, but signs reversed.

Using the definition, similarly to the ellipse, we can form the generic formula

$$\sqrt{(x-c{)}^2+y^2}-\sqrt{(x+c{)}^2+y^2}=2a$$ $$\sqrt{(x-c{)}^2+y^2}=2a+\sqrt{(x+c{)}^2+y^2}$$ $$(x-c{)}^2+y^2=4a^2+4a\sqrt{(x+c{)}^2+y^2}+(x+c{)}^2+y^2$$ $$\cancel{x^2}-2cx+\cancel{c^2}+\cancel{y^2}=4a^2+4a\sqrt{(x+c{)}^2+y^2}+\cancel{x^2}+2cx+\cancel{c^2}+\cancel{y^2}$$ $$-2cx=4a^2+4a\sqrt{(x+c{)}^2+y^2}+2cx$$ $$-4cx=4a^2+4a\sqrt{(x+c{)}^2+y^2}$$ $$-cx-a^2=a\sqrt{x^2+2cx+c^2+y^2}$$ $${\left[-(cx+a^2)\right]}^2=a^2\left[x^2+2cx+c^2+y^2\right]$$ $$c^2{x}^2+\cancel{2a^{2}cx}+a^4=a^2{x}^2+\cancel{2a^{2}cx}+a^2{c}^2+a^2{y}^2$$ $$c^2{x}^2+a^4=a^2{x}^2+a^2{c}^2+a^2{y}^2$$ $$(c^2-a^2)x^2-a^2{y}^2=(c^2-a^2)a^2$$

Now, using the identity $a^2=c^2-b^2$

$$b^2{x}^2-a^2{y}^2=b^2{a}^2$$

Divide both sides by $a^2{b}^2$

$$⟹\boxed{\frac{x^2}{a^2}-\frac{y^2}{b^2}=1}$$

..which is an equivalent expression to the one on the formula sheet!