Identify each of the conic sections:
1. x^2+xy=6
2. 3x^2+2y^2=14
3. 3x^2+3y^2-4x+3y+10=0
#1. Better review the A,B,C coefficients of the general quadratic.
hyperbola
#2. ellipse
#3. circle
Thank you so much oobleck! :)
To identify each of the conic sections, we can analyze the given equations and determine their general form. The general forms of the conic sections are as follows:
1. x^2 + xy = 6:
This equation contains both x^2 and xy terms. Conic sections that contain both squared and linear terms are classified as parabolas.
2. 3x^2 + 2y^2 = 14:
This equation contains only squared terms for both x and y, with their coefficients being positive. Conic sections with positive coefficients for both variables are classified as ellipses.
3. 3x^2 + 3y^2 - 4x + 3y + 10 = 0:
This equation contains squared terms for both x and y, along with linear terms. Additionally, it has both x and y squared terms with the same coefficient. Conic sections that have squared terms for both variables, with the same coefficient and also include linear terms, are classified as circles.
Therefore, to summarize the identification:
1. x^2 + xy = 6: Parabola
2. 3x^2 + 2y^2 = 14: Ellipse
3. 3x^2 + 3y^2 - 4x + 3y + 10 = 0: Circle