The answer explains the differences between parabolas, ellipses, circles, and hyperbolas, and provides the standard equations for hyperbolas with different orientations. It also outlines the process to find the Equations to the axes, asymptotes, and conjugate hyperbola of a hyperbola given a general form of the equation.
The difference between the equations for parabolas, ellipses, circles, and hyperbolas lies in their algebraic expressions and geometric properties. For example, parabolas have equations in the form y=ax²+bx+c, ellipses have equations in the form x²/a² + y²/b² = 1 or y²/a² + x²/b² = 1, circles have equations like (x-h)² + (y-k)² = r², and hyperbolas have equations in the form x²/a² - y²/b² = 1 or y²/b² - x²/a² = 1.
The standard equations of hyperbolas with different orientations are: Horizontal transverse axis: (x-h)²/a² - (y-k)²/b² = 1, and Vertical transverse axis: (y-k)²/a² - (x-h)²/b² = 1. The axes of a hyperbola, asymptotes, and conjugate hyperbola can be derived by manipulating the given general form of the hyperbola equation.
To find the Equations to the axes, asymptotes, and conjugate hyperbola of a hyperbola given by ax² +2hxy+by2² +2gx+2fy+c=0, you can refer the hyperbola to new coordinates, then manipulate the equation to find the desired forms. For example, replacing x with x' + xo and y with y' + yo can help simplify the equation to derive the Equations to the axes, asymptotes, and conjugate hyperbola.
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