To solve the problem of finding the equation of the hyperbola given:

1. Foci: $(-4, -3 - \sqrt{25})$ and $(-4, -3 + \sqrt{25})$,
2. Asymptotes: $y = \frac{4}{3}x + \frac{7}{3}$ and $y = -\frac{4}{3}x - \frac{25}{3}$,

We proceed as follows:

Step 1: Key characteristics of the hyperbola

• The center of the hyperbola is the midpoint of the foci: $\text{Center} = \left(-4, \frac{-3 - \sqrt{25} + (-3 + \sqrt{25})}{2}\right) = (-4, -3).$

• The slopes of the asymptotes indicate that this is a hyperbola whose axes are tilted (rotated from the standard orientation). The slopes $\pm \frac{4}{3}$ match the general form of a hyperbola's asymptotes.

Step 2: Rotation and general hyperbola equation

The general equation of a rotated hyperbola is: $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0,$ where $B \neq 0$ (since the hyperbola is rotated).

Step 3: Using asymptotes and simplifying

The asymptotes are given as: $y = \frac{4}{3}x + \frac{7}{3} \quad \text{and} \quad y = -\frac{4}{3}x - \frac{25}{3}.$

These correspond to the equations of the asymptotes: $4x - 3y + 7 = 0 \quad \text{and} \quad 4x + 3y + 25 = 0.$

The product of these asymptotes gives the equation of the hyperbola in the rotated form: $(4x - 3y + 7)(4x + 3y + 25) = 0.$

Expanding: $(4x)^2 - (3y)^2 + (4x)(3y)(-25 + 7) = 0$ Correction user math detaaill
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