Hyperbola Consider a hyperbola centered at the origin with a horizontal transverse axis. Use the definition of a hyperbola to derive its standard form
To Prove: The standard equation of a hyperbola using the definition.
Given: A hyperbola is a set of points such that their distance between the foci is constant.
Consider P(x, y) be any point on a hyperbola.
Let (a, 0) is the vertex and the points (c, 0) and(-c, 0) represents the foci of hyperbola.
Now sum of the distances from foci to the vertex is (c+a) + (a - c) = 2a, which is shown in below figure:
Now according to the definition:
Distance between two foci is constant that means the length of the major axis is 2a.
Take square both sides:
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