Animated Solution for Mathematics - Conic Sections: The equation $\frac{x^2}{1-r}-\frac{y^2}{1+r}=1,r>1$ represents

Welcome, fellow traveler on the path to JEE mastery. Today, we encounter a problem that serves as a profound reminder: in the world of mathematics, appearances can be deceiving.

We are presented with the equation:

We are given the critical constraint that $r>1$. At first glance, your intuition might scream "Hyperbola!" because of that familiar minus sign. But let us pause, breathe, and look deeper.

The beauty of coordinate geometry lies in its rigor. We cannot simply rely on visual patterns; we must dissect the components.

Look at the denominators: $(1-r)$ and $(1+r)$. The behavior of these terms is entirely dictated by the parameter $r$.

Since we are told $r>1$, we know that $(1-r)$ must be a negative value. Let us define this negative quantity as $-a^2$, where $a$ is a non-zero real constant.

Conversely, $(1+r)$ must be strictly positive (in fact, it is greater than $2$). Let us define this as $b^2$.

Now, let us perform the substitution. Our original equation:

becomes:

If we multiply this entire equation by $-1$, we arrive at a startling revelation:

Stop and reflect on this result. On the left-hand side, we have the sum of two squares, $\frac{x^2}{a^2}+\frac{y^2}{b^2}$.

Since $x^2$ and $y^2$ are always non-negative for any real $x$ and $y$, and $a^2$ and $b^2$ are positive, the left-hand side must be greater than or equal to zero. Yet, the right-hand side is $-1$.

A non-negative quantity can never equal a negative quantity. This is a mathematical impossibility. There are no real coordinates $(x,y)$ that can satisfy this equation; it represents an empty set.

This problem is a masterclass in why we must never rush. The JEE examiners love to hide contradictions behind familiar structures.

By analyzing the constraints and performing the substitution, we peeled back the mask of the "hyperbola" to reveal that no curve exists at all. The correct answer is "none of these."

Keep this vigilance in your toolkit. When you see a parameter, treat it as a variable that could change the very nature of the equation. Stay curious, stay rigorous, and keep pushing forward.