To determine the conic section represented by the equation 4x^2 + y^2 - 4x - 14y + 50 = 0, we can rewrite the equation in standard form by completing the square for both x and y terms.
Starting with x-terms: 4x^2 - 4x = 4(x^2 - x) To complete the square for x, we add and subtract (1/2)^2 = 1/4 inside the parentheses: 4(x^2 - x + 1/4) = 4(x - 1/2)^2 - 1
Now, for y-terms: y^2 - 14y = y^2 - 14y To complete the square for y, we add and subtract (14/2)^2 = 49 inside the parentheses: y^2 - 14y + 49 = (y - 7)^2
Substitute these results back into the original equation: 4(x - 1/2)^2 - 1 + (y - 7)^2 - 49 + 50 = 0 4(x - 1/2)^2 + (y - 7)^2 = 0
Since both terms are being added, this equation represents the point (x, y) = (1/2, 7) and is the equation of a point rather than a conic section. So, the given equation does not represent any conic section.
To determine the conic section represented by the equation 4x^2 + y^2 - 4x - 14y + 50 = 0, we can rewrite the equation in standard form by completing the square for both x and y terms.
Starting with x-terms:
4x^2 - 4x = 4(x^2 - x)
To complete the square for x, we add and subtract (1/2)^2 = 1/4 inside the parentheses:
4(x^2 - x + 1/4) = 4(x - 1/2)^2 - 1
Now, for y-terms:
y^2 - 14y = y^2 - 14y
To complete the square for y, we add and subtract (14/2)^2 = 49 inside the parentheses:
y^2 - 14y + 49 = (y - 7)^2
Substitute these results back into the original equation:
4(x - 1/2)^2 - 1 + (y - 7)^2 - 49 + 50 = 0
4(x - 1/2)^2 + (y - 7)^2 = 0
Since both terms are being added, this equation represents the point (x, y) = (1/2, 7) and is the equation of a point rather than a conic section. So, the given equation does not represent any conic section.