Identify the conic as a circle, parabola, ellipse, or hyperbola. $$6 x ^ { 2 } = 24 x + 2 \left( y ^ { 2 } - 8 y \right) + 20$$
A)circle
B)hyperbola
C)ellipse
D)parabola
E)none of the above

Question

Quizplus · Accepted Answer

The given equation can be rewritten in the standard form of a conic section by completing the square for both x and y terms. However, the presence of both $x^2$ and $y^2$ terms with opposite signs (after rearranging and simplifying) indicates it is a hyperbola. Specifically, the equation simplifies to a form similar to $x^2/a^2 - y^2/b^2 = 1$, characteristic of a hyperbola.

Identify the Conic as a Circle, Parabola, Ellipse, or Hyperbola 6x2=24x+2(y2−8y)+206 x ^ { 2 } = 24 x + 2 \left( y ^ { 2 } - 8 y \right) + 206x2=24x+2(y2−8y)+20

Identify the Conic as a Circle, Parabola, Ellipse, or Hyperbola $6 x ^ { 2 } = 24 x + 2 \left( y ^ { 2 } - 8 y \right) + 20$