Chapter 10 - Conics, Parametric Equations, and Polar Coordinates - Review Exercises - Page 742: 12

$$\frac{{{{\left( {x - \frac{1}{2}} \right)}^2}}}{3} - \frac{{{{\left( {y + 1} \right)}^2}}}{3} = 1$$

$$\eqalign{ & 12{x^2} - 12{y^2} - 12x + 24y - 45 = 0 \cr & {\text{Group terms}} \cr & \left( {12{x^2} - 12x} \right) - \left( {12{y^2} + 24y} \right) = 45 \cr & 12\left( {{x^2} - x} \right) - 12\left( {{y^2} + 2y} \right) = 45 \cr & {\text{Complete the square and factor}} \cr & 12\left( {{x^2} - x + \frac{1}{4}} \right) - 12\left( {{y^2} + 2y + 1} \right) = 45 + 12\left( {\frac{1}{4}} \right) - 12\left( 1 \right) \cr & 12{\left( {x - \frac{1}{2}} \right)^2} - 12{\left( {y + 1} \right)^2} = 36 \cr & {\text{Divide both sides by 36}} \cr & \frac{{{{\left( {x - \frac{1}{2}} \right)}^2}}}{3} - \frac{{{{\left( {y + 1} \right)}^2}}}{3} = 1 \cr & {\text{This equation has the form}} \cr & \frac{{{{\left( {x - h} \right)}^2}}}{{{a^2}}} - \frac{{{{\left( {y - k} \right)}^2}}}{{{b^2}}} = 1 \cr & {\text{Therefore,}} \cr & \frac{{{{\left( {x - \frac{1}{2}} \right)}^2}}}{3} - \frac{{{{\left( {y + 1} \right)}^2}}}{3} = 1 \Rightarrow \underbrace {\frac{{{{\left( {x - \frac{1}{2}} \right)}^2}}}{{{{\left( {\sqrt 3 } \right)}^2}}} - \frac{{{{\left( {y + 1} \right)}^2}}}{{{{\left( {\sqrt 3 } \right)}^2}}} = 1}_{\frac{{{{\left( {x - h} \right)}^2}}}{{{a^2}}} - \frac{{{{\left( {y - k} \right)}^2}}}{{{b^2}}} = 1} \cr & h = \frac{1}{2},{\text{ }}k = - 1,{\text{ }}a = \sqrt 3 ,{\text{ }}b = \sqrt 3 \cr & c = \sqrt {{a^2} + {b^2}} = \sqrt {3 + 3} \Rightarrow c = \sqrt 6 \cr & {\text{Characteristics of the hyperbola }} \cr & {\text{Orientation}}:{\text{ Horizontal transverse axis}} \cr & {\text{Center}}\left( {h,k} \right) \to {\text{Center}}\left( {\frac{1}{2}, - 1} \right) \cr & {\text{Vertices}}\left( {h - a,k} \right){\text{ and }}\left( {h + a,k} \right) \cr & {\text{Vertices}}\left( {\frac{1}{2} - \sqrt 3 , - 1} \right){\text{ and }}\left( {\frac{1}{2} + \sqrt 3 , - 1} \right){\text{ }} \cr & {\text{Foci}}\left( {h - c,k} \right){\text{ and }}\left( {h + c,k} \right) \cr & {\text{Foci}}\left( {\frac{1}{2} - \sqrt 6 , - 1} \right){\text{ and }}\left( {\frac{1}{2} + \sqrt 6 , - 1} \right) \cr & \underbrace {{\text{Asymptotes}}:{\text{ }}y = \pm \frac{b}{a}\left( {x - h} \right) + k}_ \Downarrow \cr & {\text{Asymptotes}}:{\text{ }}y = \pm \left( {x - \frac{1}{2}} \right) - 1 \cr & {\text{Graph}} \cr} $$