How do you think you would tell the difference between a hyperbola and a circle or ellipse in EXPANDED Form?
whatever form it is in, if the value of a variable goes to infinity in any direction and it still can satisfy the equation, it is not a circle or ellipse.
for example
x^2+y^2 = 4 (circle radius 2)
what happens if |x| or |y| > 2 ?
circle: x^2 + y^2
ellipse ax^2 + by^2
hyperbola ax^2 - by^2 or by^2 - ax^2
To determine whether a given equation in expanded form represents a hyperbola, circle, or ellipse, you need to examine the coefficients and constants present in the equation. Here is the step-by-step process:
1. Examine the terms involving x and y: A hyperbola can have the terms involving x and y squared in opposite signs (e.g., x² - y²), whereas a circle or ellipse has the same sign for both terms (e.g., x² + y² or x² - y²). In expanded form, look for these types of terms and check for the signs being either opposite or the same.
2. Check the coefficients of the x² and y² terms: In a hyperbola, the coefficients of the x² and y² terms are different, resulting in an equation with distinct squared terms like this: Ax² - By² = C. On the other hand, in a circle or ellipse, the coefficients of the x² and y² terms are the same: Ax² + By² = C.
3. Analyze the constant term: The constant term in the equation (either C or -C) has a role in determining the type of conic section. For a hyperbola, the constant terms of x and y have opposite signs (one positive, one negative). However, for a circle or ellipse, the constant term is positive.
By following these steps and analyzing the coefficients and constants, you can differentiate between a hyperbola, circle, or ellipse when given an equation in expanded form.