a) The equation represents an ellipse.
b) The equation represents a circle.
c) The equation represents a parabola.
How to infer the graphical form of a general equation
In this problem we have general equations of the form A · x² + B · y² + C · x + D · y + E = 0, which have to be modified into standard form to infer its graphical form. This procedure can be done by algebra properties:
16 · x² + 4 · y² + 96 · x - 8 · y + 84 = 0
[(4 · x)² + 2 · 12 · (4 · x)] + [(2 · y)² - 2 · 2 · (2 · y)] = - 84
[(4 · x)² + 2 · 12 · (4 · x) + 144] + [(2 · y)² - 2 · 2 · (2 · y) + 4] = 64
(4 · x + 12)² + (2 · y + 2)² = 64
4² · (x + 3)² + 2² · (y + 2)² = 64
(x + 3)² / 4 + (y + 2)² / 16 = 1 : Ellipse
x² + y² + 8 · x - 6 · y - 15 = 0
(x² + 8 · x) + (y² - 6 · y) = 15
(x² + 2 · 4 · x + 16) + (y² - 2 · 3 · y + 9) = 40
(x + 4)² + (y - 3)² = 40 : Circle
x² + 6 · x + 4 · y + 5 = 0
x² + 6 · x + 5 = - 4 · y
y = - (1 / 4) · x² - (3 / 2) · x - (5 / 4) : Parabola
To learn more on ellipses: https://brainly.com/question/19507943
#SPJ1
