Animated Solution for Mathematics - Conic Sections: Let $S=\{(x,y)\in R^2:\frac{y^2}{1+r}-\frac{x^2}{1-r}=1\}$, where $r\ne \pm 1$. Then S represents :

The given equation is:

At first glance, this appears to be a hyperbola. However, the parameter $r$ acts as a shapeshifter, determining the fundamental nature of the conic section based on its value.

When $r>1$, the term $(1+r)$ is positive, but the term $(1-r)$ is negative. To analyze the geometry, we rewrite $(1-r)$ as $-(r-1)$, where $(r-1)$ is a positive quantity.

Substituting this into the original equation yields:

The two negative signs cancel, transforming the equation into:

This is the standard form of an ellipse, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$.

We identify the parameters as $a^2=r-1$ and $b^2=r+1$. Since $r+1>r-1$, the denominator under $y^2$ is larger, indicating that the ellipse is stretched along the $y$-axis.

This confirms that the curve is a vertical ellipse.

To find the eccentricity $e$, we use the standard formula for a vertical ellipse:

Substituting our specific values for $a^2$ and $b^2$, we obtain:

We combine the terms under a common denominator of $(r+1)$:

Simplifying the numerator, the $r$ terms cancel out:

The final eccentricity of the ellipse is:

This result confirms that for $r>1$, the curve is indeed an ellipse. The journey of transformation demonstrates how algebraic constraints dictate the geometric soul of the conic section.