To find the equation of a rectangular hyperbola passing through the points $(-1.82, 0.46958)$ and $(-1.51, -0.22238)$, we start with the general form of the equation you mentioned:

$y = \pm \frac{a}{x \pm b} \pm c$

Let's solve for the values of $a$, $b$, and $c$ by substituting the given coordinates.

Step 1: Substituting the points

Substitute the coordinates $(-1.82, 0.46958)$ and $(-1.51, -0.22238)$ into the general equation.

For the first point $(-1.82, 0.46958)$:

$0.46958 = \pm \frac{a}{-1.82 \pm b} \pm c$

For the second point $(-1.51, -0.22238)$:

$-0.22238 = \pm \frac{a}{-1.51 \pm b} \pm c$

Step 2: Choosing signs and solving for $a$, $b$, and $c$

To determine the correct values of $a$, $b$, and $c$, we need to solve these equations simultaneously.

Given the complexity, it might be simpler to start by making reasonable assumptions about the signs and then solving for the unknowns.

1. Assume $y = \frac{a}{x + b} + c$.
2. Substitute the coordinates into the equations and solve for $a$, $b$, and $c$.

I'll now calculate these values.The solutions for $a$, $b$, and $c$ are expressed in terms of $c$:

1. $a = 0.448002774726863c^2 - 0.11074628591248c - 0.0467827795306087$
2. $b = 1.72037314295624 - 0.448002774726863c$
3. $c = c$ (where $c$ is a free parameter)

This means the equation of the hyperbola depends on the value chosen for $c$. For any specific $c$, you can determine the corresponding values of $a$ and $b$, and hence the specific equation of the hyperbola.

Would you like to choose a value for $c$ to see the specific equation, or do you have any further questions?

Related Questions:

1. How would changing the value of $c$ affect the shape of the hyperbola?
2. What if we used the alternative sign choices in the original equation?
3. How can we verify if a specific $c$ value satisfies both points?
4. What are the asymptotes of this hyperbola based on the derived equation?
5. How does this hyperbola compare to one with a different set of parameters?

Tip:

Remember that $c$ influences the vertical shift of the hyperbola, while $a$ and $b$ shape the curve's width and horizontal translation.