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) 8x2  2y2  3x + 2y  6=0 

b) 6y2 + 4x  12y  24=0

c) 9x2  4y2  18x + 12y=0

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Expert Answer

Demi-Leigh Barrera
Answered 2020-11-09 Author has 97 answers

Step 1

a) Consider the equation 8x2  2y2  3x + 2y  6=0 It is known that the equation of conic section is in the form Ax2 + Cy2 + Dx + Ey + F=0 In the given equation, the coefficient of x2 and y2 are A=8 and C= 2, respectively, which means AC= 16, that is, 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 6y2 + 4x  12y  24=0. It is known that the equation of conic section is in the form Ax2 + Cy2 + Dx + Ey + F=0. Here, the given equation does not have the term of x2, 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 9x2  4y2  18x + 12y=0. It is known that the equation of conic section is in the form Ax2 + Cy2 + Dx + Ey + F=0. In the given equation, the coefficient of x2 and y2 are A= 9 and C= 4, respectively, which means AC=36, that is, AC > 0. Therefore, this is the equation of an ellipse. Hence, the conic section of the given equation is an ellipse.

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