The type of conic sections for the nondegenerate equations given below.a) 6x^{2} + 3x + 10y=10y^{2} + 8b) 3x^{2} + 18xy=5x + 2y + 9c) 4x^{2} + 8x - 5= -y^{2} + 6y + 3

floymdiT 2020-10-18 Answered

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

a) 6x2 + 3x + 10y=10y2 + 8

b) 3x2 + 18xy=5x + 2y + 9

c) 4x2 + 8x  5= y2 + 6y + 3

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Expert Answer

Bella
Answered 2020-10-19 Author has 81 answers

a) Consider the equation 6x2 + 3x + 10y=10y2 + 8.

This equation can be written as: 6x2  10y2 + 10y  8=0.

It is known that the equation is in the form of Ax2 + Cy2 + Dx + Ey + F=0

Here, the coefficient of x2 and y2 are A=6 and C= 10, which means AC= 60, that is, AC < 0.

Therefore, the equation is of a hyperbola. Hence, the conic section of the given equation is a hyperbola.

b) Consider the equation 3x2 + 18xy=5x + 2y + 9.

This equation caan be written as: 3x2 + 18xy  5x  2y  9=0, which is in the form of AX2 + Bxy + Cy2 + Dx + Ey + F=0.

Solve for B2  4AC
B2  4AC=(18)  4(3)(0)
=324  0
=324

This shows that B2  4AC > 0. Therefore, the equation is of a hyperbola. Hence, the conic section of the given equation is a heperbola.

c) Consider the equation 4x2 + 8x  5= y2 + 6y + 3. T

his equation can written as: 4x2 + y2 + 8x  6y  8=0. It is known that the equation is in the form AX2 + Cy2 + Dx + Ey + F=0 Here, A=4 and C=1, which means AC=4 and AC > 0.

Therefore, the equation is of an ellipse. Hence, the conic section of the given equation is an ellipse.

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