 # Identify the equation without applying a rotation of axes. displaystyle{4}{x}^{2}-{8}{x}{y}+{4}{y}^{2}-{3}{x}+{6}={0} Brittney Lord 2021-02-25 Answered
Identify the equation without applying a rotation of axes.
$4{x}^{2}-8xy+4{y}^{2}-3x+6=0$
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Step 1
Consider the provided equation:
$4{x}^{2}-8xy+4{y}^{2}-3x+6=0$
Step 2
Now, compare this equation with the general form of conic section:
$A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0$
Step 3
Now, it is known that:
If ${B}^{2}-4AC<0$, then it will be either circle or an ellipse.
If ${B}^{2}-4AC=0$, then it will be a parabola.
If ${B}^{2}-4AC>0$, then it will be a hyperbola.
Now, here we have, $A=4,B=-8\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}C=4$
So, ${B}^{2}-4AC={\left(-8\right)}^{2}-4\left(4\right)\left(4\right)=64-64=0$
Since, ${B}^{2}-4AC=0$
Hence, the provided equation represents a parabola if a conic section exists.