Question

What conic does the equation 6x2-5xy-6y2+14x+5y+4=0represent?

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{6x^2-5xy-6y^2+14x+5y+4=0}$

$\underline{\textbf{To find:}}$

$\textsf{What conic does the equation represent?}$

$\underline{\textbf{Solution:}}$

$\underline{\textbf{General equation of conic:}}$

$\mathsf{A\,x^2+B\,xy+C\,y^2+D\,x+E\,y+F=0}$

$\mathsf{Consider,}$

$\mathsf{6x^2-5xy-6y^2+14x+5y+4=0}$

$\textsf{Comparing this equation with}\,\mathsf{A\,x^2+B\,xy+C\,y^2+D\,x+E\,y+F=0}$

$\mathsf{we\;get}$

$\mathsf{A=6,\;B=-5,\;C=-6}$

$\mathsf{B^2-4\,AC=(-5)^2-4(6)(-6)}$

$\mathsf{B^2-4\,AC=25+144}$

$\mathsf{B^2-4\,AC=169\,>\,0}$

$\implies\mathsf{B^2-4\,AC\;>\,0}$

$\therefore\textsf{The given equation represents a}\;\textbf{Hyperbola}$