The conic section whose equation is x^2 - 3y^2 - 8x + 12y + 16 = 0, is in Position is I. Is this statement true or false
To determine the position of the conic section whose equation is given, let's first rewrite the equation in standard form:
x^2 - 3y^2 - 8x + 12y + 16 = 0
Rearranging the terms, we have:
x^2 - 8x - 3y^2 + 12y + 16 = 0
Completing the square for x terms:
(x^2 - 8x + 16) - 3y^2 + 12y + 16 = 16
(x - 4)^2 - 3y^2 + 12y + 32 = 16
(x - 4)^2 - 3y^2 + 12y + 32 - 16 = 0
(x - 4)^2 - 3y^2 + 12y + 16 = 0
Now, let's analyze the equation to determine the type and position of the conic section.
The given equation has both x^2 and y^2 terms, with opposite coefficients (-3y^2 and +1x^2). This indicates that it is an equation of a hyperbola.
To further determine its position, we need to calculate the discriminant (b^2 - 4ac) of the y^2 term. In this case, a = -3, b = 12, and c = 16.
Discriminant = (12)^2 - 4(-3)(16) = 144 + 192 = 336
Since the discriminant is positive (336 > 0), the conic section is a hyperbola that is positioned in Quadrant I.
Therefore, the statement "The conic section whose equation is x^2 - 3y^2 - 8x + 12y + 16 = 0 is in Position I" is true.