Determine whether the statement is true or false? Justify your answer.
The graph of the equation $x ^ { 2 } + x y + k y ^ { 2 } + 12 x + 16 = 0$ where, $k$ is any constant less than $\frac { 1 } { 4 }$ ,is a hyperbola.

A)False.For the graph to be a hyperbola,the discriminant must be less than zero.
B)True.For the graph to be a hyperbola,the discriminant must be greater than zero.

Question

Determine whether the statement is true or false? Justify your answer. ​
The graph of the equation $x ^ { 2 } + x y + k y ^ { 2 } + 12 x + 16 = 0$ where, $k$ is any constant less than $\frac { 1 } { 4 }$ ,is a hyperbola.
​
A)False.For the graph to be a hyperbola,the discriminant must be less than zero.
B)True.For the graph to be a hyperbola,the discriminant must be greater than zero.

Quizplus · Accepted Answer

The discriminant for a conic section $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ is $B^2 - 4AC$. For a hyperbola, this discriminant must be greater than zero. Here, $A = 1$, $B = 1$, and $C = k$, so the discriminant is $1^2 - 4(1)(k) = 1 - 4k$. For $k < \frac{1}{4}$, $1 - 4k > 0$, confirming it is a hyperbola.

Determine Whether the Statement Is True or False? Justify Your x2+xy+ky2+12x+16=0x ^ { 2 } + x y + k y ^ { 2 } + 12 x + 16 = 0x2+xy+ky2+12x+16=0

Determine Whether the Statement Is True or False? Justify Your $x ^ { 2 } + x y + k y ^ { 2 } + 12 x + 16 = 0$