In the realm of geometry, particularly conic sections, understanding the position of a point in relation to a hyperbola is a significant concept. In this discussion, we will delve into the three scenarios where a point P, denoted as P(x1, y1), can be situated either inside, outside or on the hyperbola. Let's consider a hyperbola represented by the equation (x2/a2) - (y2/b2) = 1.
The diagram below offers a clear visual representation of the interior and exterior of a hyperbola.
Step 1: Start by writing down the equation of the hyperbola in this format:
S = (x2/a2) - (y2/b2) - 1
Step 2: Next, substitute the point (x1, y1) into the equation S.
Which gives us: S1 = (x12/a2) - (y12/b2) - 1
Step 3: Now, based on the value of S1, we can determine the position of the point P(x1, y1).
Scenario 1: If S1 > 0, the point P(x1, y1) lies inside the hyperbola.
Scenario 2: If S1 = 0, the point P(x1, y1) lies on the hyperbola.
Scenario 3: If S1 < 0, the point P(x1, y1) lies outside the hyperbola.
Introduction to Conic Sections
Example 1:
Determine the position of the point (3, -4) in relation to the hyperbola x2/16 - y2/9 = 1.
Example 2:
Identify the position of the point (6, -2) with respect to the hyperbola 4x2 - 9y2 = 36.
Give the standard equation of a hyperbola.
The standard equation of a hyperbola is (x^2/a^2) – (y^2/b^2) = 1.
How to check that a point (x1, y1) lies on a hyperbola?
Substitute (x1, y1) in the equation S = (x^2/a^2) – (y^2/b^2) – 1. If S = 0, then the point (x1, y1) lies on the hyperbola.
What is the value of S at any exterior point of the hyperbola, if S = (x^2/a^2) – (y^2/b^2) – 1?
The value of S at any exterior point of the hyperbola is negative.
How to check that a point (x1, y1) lies inside a hyperbola?
Substitute (x1, y1) in the equation S = (x^2/a^2) – (y^2/b^2) – 1. If S > 0, then the point (x1, y1) lies inside the hyperbola.