Determine if the graph of the equation below is a parabola, circle, ellipse, hyperbola, point, line, two lines, or empty. Enter the most specific answer.
y^2 - x +5y - 25 = x^2 - 6x + 1
+128474 y^2 - x + 5y - 25 = x^2 - 6x + 1 rewrite as
y^2 + 5y - x^2 + 5x - 26 = 0
-1x^2 + 0xy + 1y^2 + 5x + 5y - 26 = 0
General (Standard Form) Equation Of A Conic Section
Ax^2+Bxy+Cy^2+Dx+Ey+F=0,where A,B,C,D,E,F are constants
From the standard equation, it is easy to determine the conic type eg
B^2−4AC<0 , if a conic exists, then it is a circle or ellipse
B^2−4AC=0, if a conic exists, then it is a parabola
B^2−4AC>0 , if a conic exists, it is a hyperbola
We have :
0^2 - 4 (-1) (1) = 4 > 0
This is a hyperbola that opens upward / downward
