 # The type of conic sections for the nondegenerate equations given below.a) 8x^{2} - 2y^{2} - 3x + 2y - 6=0b) -6y^{2} + 4x - 12y - 24=0c) -9x^{2} - 4y^{2} - 18x + 12y=0 Lennie Carroll 2020-11-08 Answered

The type of conic sections for the nondegenerate equations given below.

a) $$\displaystyle{8}{x}^{{{2}}}\ -\ {2}{y}^{{{2}}}\ -\ {3}{x}\ +\ {2}{y}\ -\ {6}={0} \$$

b) $$\displaystyle-{6}{y}^{{{2}}}\ +\ {4}{x}\ -\ {12}{y}\ -\ {24}={0}$$

c) $$\displaystyle-{9}{x}^{{{2}}}\ -\ {4}{y}^{{{2}}}\ -\ {18}{x}\ +\ {12}{y}={0}$$

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Step 1

a) Consider the equation $$\displaystyle{8}{x}^{{{2}}}\ -\ {2}{y}^{{{2}}}\ -\ {3}{x}\ +\ {2}{y}\ -\ {6}={0}$$ It is known that the equation of conic section is in the form $$\displaystyle{A}{x}^{{{2}}}\ +\ {C}{y}^{{{2}}}\ +\ {D}{x}\ +\ {E}{y}\ +\ {F}={0}$$ In the given equation, the coefficient of $$\displaystyle{x}^{{{2}}}\ \text{and}\ {y}^{{{2}}}\ \text{are}\ {A}={8}\ \text{and}\ {C}=\ -{2}$$, respectively, which means $$\displaystyle{A}{C}=\ -{16},$$ that is, $$\displaystyle{A}{C}\ {<}\ {0}$$. Therefore, this is the equation of a hyperbola. Hence, the conic section of the given equation is a hyperbola.

Step 2

b) Consider the equation $$\displaystyle-{6}{y}^{{{2}}}\ +\ {4}{x}\ -\ {12}{y}\ -\ {24}={0}.$$ It is known that the equation of conic section is in the form $$\displaystyle{A}{x}^{{{2}}}\ +\ {C}{y}^{{{2}}}\ +\ {D}{x}\ +\ {E}{y}\ +\ {F}={0}.$$ Here, the given equation does not have the term of $$\displaystyle{x}^{{{2}}}$$, which means $$\displaystyle{A}={0},$$ that is, $$\displaystyle{A}{C}={0}$$. Therefore, this is the equation of parabola. Hence, the conic section of the given equation is a parabola. Step 3 c) Consider the equation $$\displaystyle-{9}{x}^{{{2}}}\ -\ {4}{y}^{{{2}}}\ -\ {18}{x}\ +\ {12}{y}={0}.$$ It is known that the equation of conic section is in the form $$\displaystyle{A}{x}^{{{2}}}\ +\ {C}{y}^{{{2}}}\ +\ {D}{x}\ +\ {E}{y}\ +\ {F}={0}$$. In the given equation, the coefficient of $$\displaystyle{x}^{{{2}}}$$ and $$\displaystyle{y}^{{{2}}}$$ are $$\displaystyle{A}=\ -{9}$$ and $$\displaystyle{C}=\ -{4},$$ respectively, which means $$\displaystyle{A}{C}={36},$$ that is, $$\displaystyle{A}{C}\ {>}\ {0}$$. Therefore, this is the equation of an ellipse. Hence, the conic section of the given equation is an ellipse.