How do use the discriminant test to determine whether the graph #x^2+5xy+1-y^2-16=0# whether the graph is parabola, ellipse, or hyperbola?

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Answer 1 · 2018-07-20T06:57:29

It is a hyperbola.

Let the equation be of the type #Ax^2+Bxy+Cy^2+Dx+Ey+F=0#

then if

#B^2-4AC=0# and #A=0# or #C=0#, it is a parabola

#B^2-4AC<0# and #A=C#, it is a circle

#B^2-4AC<0# and #A!=C#, it is an ellipse

#B^2-4AC>0#, it is a hyperbola

In the given equation #x^2+5xy+1-y^2-16=0# or #x^2+5xy-y^2-15=0#

#A=1#, #B=5# and #C=-1#

Therefore, #B^2-4AC=25+4=29#

Hence, it is a hyperbola.

graph{x^2+5xy+1-y^2-16=0 [-20, 20, -10, 10]}