# 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)

b)

c)

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Step 1

a) Consider the equation It is known that the equation of conic section is in the form In the given equation, the coefficient of , respectively, which means that is, . 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 It is known that the equation of conic section is in the form Here, the given equation does not have the term of ${x}^{2}$, which means $A=0,$ that is, $AC=0$. Therefore, this is the equation of parabola. Hence, the conic section of the given equation is a parabola. Step 3 c) Consider the equation It is known that the equation of conic section is in the form . In the given equation, the coefficient of ${x}^{2}$ and ${y}^{2}$ are and respectively, which means $AC=36,$ that is, . Therefore, this is the equation of an ellipse. Hence, the conic section of the given equation is an ellipse.