Animated Solution for Mathematics - Conic Sections: Let $f(x)=x^2+9,g(x)=\frac{x}{x-9}$ and $a=f\circ g(10),b=g\circ f(3)$. If $e$ and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac{x^2}{a}+\frac{y^2}{b}=1$, then $8e^2+l^2$ is equal to.

Welcome, fellow traveler on the JEE journey. Today, we are not just solving a problem; we are peeling back the layers of a mathematical onion.

We have a composite function problem that leads us into the elegant world of conic sections. It might look like a simple algebra exercise at first, but it requires precision and a deep understanding of how functions and geometry dance together.

We start with two functions: $f(x)=x^2+9$ and $g(x)=\frac{x}{x-9}$. Our mission is to find two constants, $a$ and $b$, which will define our ellipse.

Think of composite functions like a relay race. In $a=f(g(10))$, the number $10$ is the runner. It first passes through the $g$ function, and the result of that sprint is then handed off to the $f$ function.

Let us evaluate the inner function first:

Now, we take this result and feed it into $f(x)$: Our first constant is locked in: $a=109$.

Now, for $b=g(f(3))$. We start with:

Now, we pass this $18$ into $g(x)$: We have successfully navigated the algebra: $a=109$ and $b=2$.

With $a$ and $b$ in hand, our ellipse equation becomes:

This is a classic horizontal ellipse. Because $109>2$, the major axis is clearly along the $x$-axis. In the language of conic sections, the semi-major axis is $\sqrt{109}$ and the semi-minor axis is $\sqrt{2}$.

Now, we need the eccentricity $e$. The formula for a horizontal ellipse is $e^2=1-\frac{b}{a}$. Substituting our values:

Next, the latus rectum $l$. The standard length is $\frac{2\times (\text{semi-minor}{)}^2}{\text{semi-major}}$. With our values:

The question asks for $l^2$. So, we square this value:

We have arrived at the final hurdle: $8e^2+l^2$. We have our components ready: $e^2=\frac{107}{109}$ and $l^2=\frac{16}{109}$.

Let us assemble them:

Multiplying $8$ by $107$ gives us $856$. So, we have:

If you perform the division, you will find that $109\times 8=872$. The final result is exactly 8.