The conic section whose equation is x 2 - 3y 2 - 8x + 12y + 16 = 0, is in Position is I. Is this true or false?
To determine the position of a conic section, we need to examine the coefficients of the equation. In this case, the given equation is:
x^2 - 3y^2 - 8x + 12y + 16 = 0
To analyze the position, we focus on the quadratic terms (x^2 and y^2) and their coefficients.
The coefficient of x^2 is 1, while the coefficient of y^2 is -3.
Since the coefficients of x^2 and y^2 are of opposite signs, we can conclude that the conic section is a hyperbola.
Now, let's determine the specific position of the hyperbola.
The general equation of a hyperbola in standard form is:
(x - h)^2 / a^2 - (y - k)^2 / b^2 = 1
Comparing this general form to the given equation, we can rewrite the equation as:
(x^2 - 8x + 16) / 9 - (y^2 + 12y + 36) / 3 = 1
By factoring the denominators, we obtain:
[(x - 4)^2 / 3^2] - [(y + 6)^2 / (√3)^2] = 1
Comparing this equation to the standard form, we can determine that h = 4, k = -6, a = 3, and b = √3.
Using these values, we can conclude that the given hyperbola is in Position I. Therefore, the statement is true.