Part 3/5:
Apply the Distance Formula: The distances (D1) and (D2) can be written using the distance formula. This involves utilizing the coordinates of the foci and the point on the hyperbola.
Set Up the Equation: From the definition:
[ |D1 - D2| = 2a ]
We can square both (D1) and (D2) to eliminate the square roots that arise from the distance calculations.
Manipulate Algebraically: After substituting and simplifying the expressions, you end up rearranging the terms to achieve the standard form of the hyperbola’s equation.
Conclude with the Equation: Through this manipulation, you arrive at the relationship that defines the hyperbola in the standard form.