identify the conic sections below (circle,hyperbola,parabola,ellipse).

Question

identify the conic sections below (circle,hyperbola,parabola,ellipse).

a)3x2+3y2-2y=4
b)3x2-9y2+2x-4y=7
c)2x2+5y2-7x+3y-4=0
d)3y2-4x+17y=-10

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3 answers
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Answer 1

MathMate answered
7 years ago

hints:

1. Circles have equal coefficients of x² and y²
e.g. 2x²+2y²+3x-5y=17
2. hyperbola have different sign coefficients of x² and y²
e.g. 4x²-8y²-2x+5y=24
3. Parabolas have one of the squared terms missing,
e.g. x²-4x+4=0, or 2y²-4x+4=0
4. Ellipses have same sign coefficients of x² and y², but unequal numerically.
e.g. 2x²+4x+2y²-6y-46=0

(a) is an ellipse because both squared terms have a coefficient of 3.
(b)...(d) will be left as exercise for you.
Feel free to post your answers for checking. Please include justification.

Answer 2

Step-by-Step Bot answered
1 year ago

a) To identify the conic section in the equation 3x^2 + 3y^2 - 2y = 4, we can start by rearranging it in standard form:

3x^2 + 3y^2 - 2y - 4 = 0

Next, let's divide all the terms by 4:
(3/4)x^2 + (3/4)y^2 - (1/2)y - 1 = 0

Since the coefficients of x^2 and y^2 are the same, and they have different signs, this equation represents an ellipse.

b) To identify the conic section in the equation 3x^2 - 9y^2 + 2x - 4y = 7, we can start by rearranging it in standard form:

3x^2 + 2x - 9y^2 - 4y - 7 = 0

Next, let's divide all the terms by 7:
(3/7)x^2 + (2/7)x - (9/7)y^2 - (4/7)y - 1 = 0

Since the coefficients of x^2 and y^2 have different signs, this equation represents a hyperbola.

c) To identify the conic section in the equation 2x^2 + 5y^2 - 7x + 3y - 4 = 0, we can start by rearranging it in standard form:

2x^2 - 7x + 5y^2 + 3y - 4 = 0

Next, let's group the x terms together and the y terms together:
(2x^2 - 7x) + (5y^2 + 3y) - 4 = 0

Since the coefficients of x^2 and y^2 have the same sign, this equation represents an ellipse.

d) To identify the conic section in the equation 3y^2 - 4x + 17y = -10, we can start by rearranging it in standard form:

-4x + 3y^2 + 17y + 10 = 0

Next, let's group the x term and rearrange the equation:
-4x = -3y^2 - 17y - 10

Since x is only present as a linear term and not squared, this equation represents a parabola.

Answer 3

Explain Bot answered
1 year ago

To identify the conic sections, we need to analyze the equations of the given equations.

a) 3x^2 + 3y^2 - 2y = 4

We can rewrite this equation as:
3x^2 + 3y^2 - 2y - 4= 0

This equation represents an ellipse. To identify it as an ellipse, we need to ensure that the coefficients of x^2 and y^2 terms are equal and have the same sign (positive), which is the case with this equation.

b) 3x^2 - 9y^2 + 2x - 4y = 7

We can rewrite this equation as:
3x^2 + 2x - 9y^2 - 4y - 7 = 0

This equation represents a hyperbola. To identify it as a hyperbola, we need to ensure that the coefficients of x^2 and y^2 terms are equal but with opposite signs. In this equation, the coefficient of x^2 is positive (3), and the coefficient of y^2 is negative (-9), satisfying this condition.

c) 2x^2 + 5y^2 - 7x + 3y - 4 = 0

This equation represents an ellipse. Similar to equation a), we need the coefficients of x^2 and y^2 terms to be equal and have the same sign, which is true here.

d) 3y^2 - 4x + 17y = -10

We can rewrite this equation as:
3y^2 + 17y - 4x + 10 = 0

This equation represents a parabola. To identify it as a parabola, we need only one squared variable and no coefficient for the linear term. In this equation, we have y^2 as the squared variable, and there is no coefficient for the x term, satisfying this condition.

In summary:
a) Ellipse
b) Hyperbola
c) Ellipse
d) Parabola