Question 1142038
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Your conic section is presented by equation


    4x² + 14xy + 5y² + 18x − 6y + 30 = 0.


The equation has a mixed term xy -- therefore, it may happen that the standard high school knowledge is not enough 
to answer the question.


But the answer does exist in Analytic Geometry.


To determine the type of the conic section, calculate the <U>discriminant</U> 


    d = B^2 - 4*A*C,


where  B = 14 is the coefficient at the "xy"-term;

       A = 4  is the coefficient at the "x^2"-term;

       C = 5  is the coefficient at the "y^2"-term.


In your case  the discriminant  d = {{{14^2 - 4*4*5}}} =  116  is a positive real number.


It means that the conic section is a HYPERBOLA.


For details, read your textbook or learn from the universal source WIKIPEDIA - see Wikipedia article  

    <A HREF=https://en.wikipedia.org/wiki/Conic_section>https://en.wikipedia.org/wiki/Conic_section</A>

    https://en.wikipedia.org/wiki/Conic_section
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