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
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.
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.
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