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

Step 1

a) Consider the equation ${\displaystyle 8x^2-2y^2-3x+2y-6=0}$ It is known that the equation of conic section is in the form ${\displaystyle A{x}^2+C{y}^2+Dx+Ey+F=0}$ 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 ${\displaystyle AC=-16,}$ that is, ${\displaystyle AC<0}$. 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 ${\displaystyle -6y^2+4x-12y-24=0.}$ It is known that the equation of conic section is in the form ${\displaystyle A{x}^2+C{y}^2+Dx+Ey+F=0.}$ Here, the given equation does not have the term of ${\displaystyle x^2}$, which means ${\displaystyle A=0,}$ that is, ${\displaystyle AC=0}$. Therefore, this is the equation of parabola. Hence, the conic section of the given equation is a parabola. Step 3 c) Consider the equation ${\displaystyle -9x^2-4y^2-18x+12y=0.}$ It is known that the equation of conic section is in the form ${\displaystyle A{x}^2+C{y}^2+Dx+Ey+F=0}$. 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 ${\displaystyle AC=36,}$ that is, ${\displaystyle AC>0}$. Therefore, this is the equation of an ellipse. Hence, the conic section of the given equation is an ellipse.