Question

The greatest slope along the graph represented by the equation $4x^2-y^2+2y-1=0$, is

• A. -3
• B. -2
• C. 2
• D. 3

Answer 1

Answer: (C)

2

Answer 2

The given equation $4x^2 - y^2 + 2y - 1 = 0$ can be rewritten by completing the square for the $y$ terms: $4x^2 - (y^2 - 2y + 1) = 0$, which simplifies to $4x^2 - (y-1)^2 = 0$. This is a difference of squares: $(2x)^2 - (y-1)^2 = 0$. Factoring this expression yields $(2x - (y-1))(2x + (y-1)) = 0$, which further simplifies to $(2x - y + 1)(2x + y - 1) = 0$. This equation represents a pair of straight lines: $2x - y + 1 = 0$ and $2x + y - 1 = 0$. Rearranging these equations to solve for $y$, we get $y = 2x + 1$ and $y = -2x + 1$. The slopes of these lines are $m_1 = 2$ and $m_2 = -2$, respectively. The greatest slope along the graph is the maximum of these two values, which is $\max(2, -2) = 2$.