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A hyperbola is a type of conic section that can be defined by its geometric properties. To derive the equation of a hyperbola, we start with its standard form. For a hyperbola centered at the origin with its foci on the x-axis, the equation is \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\). Here, \(a\) and \(b\) are the distances that define the shape and size of the hyperbola.
We need to determine the values of \(a^2\) and \(b^2\) to complete the equation. These values depend on specific points through which the hyperbola passes. In our problem, these points are \((4, -2)\) and \((7, -6)\)
By substituting these points into the general equation, we create a system of equations. Solving this system allows us to identify the specific values for \(a^2\) and \(b^2\).
Thus, we can accurately construct the equation of the hyperbola.

Analytic geometry helps us understand and represent geometric objects like hyperbolas using algebra. The process includes several clear steps:

• Identify the General Equation: For a hyperbola centered at the origin with its foci on the x-axis, we use \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\).
• Substitute Given Points: Using points \((4, -2)\) and \((7, -6)\), we plug these coordinates into our equation to form a system.
• Solve the System of Equations: This system helps us find specific values for \(a^2\) and \(b^2\).
• Simplify and Solve: Subtract one equation from the other to eliminate variables and solve for \(a^2\).
• Determine \(b^2\) : Use the relation between \(a^2\) and \(b^2\) to find \(b^2\).
• Write the Final Equation: Substitute the values back into the general equation to get the specific hyperbola equation.

This step-by-step approach ensures accuracy and clarity in deriving the equation.

In coordinate geometry, solving systems of equations is a method to find the exact parameters of geometric figures, like a hyperbola. Here’s a detailed breakdown:
Start by forming equations from given conditions. For the hyperbola, we use the points \((4, -2)\) and \((7, -6)\) in the standard equation \( \frac{16}{a^2} - \frac{4}{b^2} = 1 \) and \( \frac{49}{a^2} - \frac{36}{b^2} = 1 \). These create a system of linear equations.
Simplify the System: Organize and simplify these equations to make solving easier.

Solve Step-by-Step:

• Subtract one equation from the other to eliminate common terms and find one variable.
• Solve for \(a^2\) first.
• Once \(a^2\) is found, substitute it back to find \(b^2\).

By solving these equations accurately, we’re ensuring that the final values of \(a^2\) and \(b^2\) represent a precise description of the hyperbola. This method is essential for solving more complex coordinate geometry problems too.