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.