How do i know if a conic section is a circle, ellipse, hyperbola, or parabola? based on it's equation such as 4x^2 - y^2 + 24x + 4y + 28 = 0
General equation:
ax^2 + by^2 + cx + dy + e = 0
1) if a = b ---> circle
2) if both a and b are positive, but different, ---> ellipse
3) if a is positive and b is negative, ----> hyperbola
4) if either a or b, but not both, are zero ----> parabola
5) if both a and b are zero ---> straight line
so yours is a hyperbola
Find the transformed equation of the hyperbola xy = 4 when rotated 45?
Thanks reiny. This helped a lot.
To determine the type of conic section based on its equation, you need to bring the equation into a standardized form. In this case, let's work with the given equation:
4x^2 - y^2 + 24x + 4y + 28 = 0
1. Rearrange the equation by combining like terms:
4x^2 + 24x - y^2 + 4y + 28 = 0
2. Group the x-terms together and the y-terms together:
(4x^2 + 24x) + (-y^2 + 4y) + 28 = 0
3. Complete the square for both the x-terms and y-terms separately:
To complete the square for the x-terms, divide the coefficient of x (24) by 2, square it, and add it to both sides of the equation:
(4x^2 + 24x + 36) + (-y^2 + 4y) + 28 = 36
To complete the square for the y-terms, divide the coefficient of y (4) by 2, square it, and add it to both sides of the equation:
(4x^2 + 24x + 36) + (-y^2 + 4y + 4) + 28 = 36 + 4
Simplify:
(4x + 6)^2 - (y - 2)^2 + 28 = 40
4. Rearrange the equation to isolate the squared terms:
(4x + 6)^2 - (y - 2)^2 = 40 - 28
(4x + 6)^2 - (y - 2)^2 = 12
Now we can analyze the equation to determine the conic section:
- If the squared terms have the same sign and the coefficients with them are equal (as they are both 1 in this case), it is an ellipse.
- If the squared terms have opposite signs (like in this equation), it is a hyperbola.
- If one squared term is missing (i.e., with coefficient 0), it is a parabola.
- If the coefficients of both squared terms are the same but nonzero, it is a circle.
In our given equation, (4x + 6)^2 - (y - 2)^2 = 12, the equation represents a hyperbola because the squared terms have opposite signs.