Animated Solution for Mathematics - Conic Sections: The equation of the chord, of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ whose mid-point is $(3,1)$ is:

Unlocking the Chord

Welcome, future engineers. Today, we are not just solving a problem; we are uncovering a hidden symmetry in the world of conic sections.

Imagine you are standing before the ellipse defined by the equation:

It is a beautiful, smooth curve, stretched along the $x$-axis. Now, imagine a line cutting through this ellipse, creating a chord. We are told that this chord has a very specific property: its midpoint is fixed at the coordinate $(3,1)$. Our mission is to find the equation of this line.

Many students, when faced with this, immediately reach for the slope-point form: $y-y_1=m(x-x_1)$. They try to find the slope $m$ by substituting the line into the ellipse equation and forcing the roots to be symmetric.

While that method is mathematically sound, it is a trap. It is a path filled with quadratic equations, discriminant calculations, and a high probability of arithmetic errors.

In the JEE Advanced, time is your most precious currency. We need a more elegant approach. We need the power of the $T=S_1$ theorem.

There is a profound result in coordinate geometry that states: for any conic section, the equation of a chord bisected at a point $(x_1,y_1)$ is given by $T=S_1$.

Think of $T$ as the 'tangent-like' expression. If you were to write the equation of a tangent at $(x_1,y_1)$, you would replace $x^2$ with $x{x}_1$ and $y^2$ with $y{y}_1$. That is exactly what we do here.

And $S_1$? That is simply the 'power of the point'—the value you get when you plug the coordinates $(x_1,y_1)$ directly into the equation of the curve.

Let us apply this to our ellipse. Our midpoint is $(x_1,y_1)=(3,1)$.

First, we construct $T$. We take our ellipse equation $\frac{x^2}{25}+\frac{y^2}{16}=1$ and perform the transformation $x^2\to x{x}_1$ and $y^2\to y{y}_1$. This gives us:

Next, we calculate $S_1$. This is the value of the ellipse equation at our midpoint $(3,1)$:

To add these fractions, we find the common denominator, which is $25\times 16=400$. Thus, we have:

Now, we equate them. The magic of the theorem tells us that $T=S_1$ is the equation of our chord:

To make this look like the standard linear equations we see in our options, we clear the denominators by multiplying the entire equation by $400$:

This simplifies beautifully:

Look at that result: $48x+25y=169$. It is clean, it is precise, and it was achieved without solving a single complex quadratic.

Remember this: whenever you see a problem involving a chord and its midpoint, do not hesitate. Invoke the $T=S_1$ theorem. It is not just a formula; it is a testament to the symmetry of the conic sections. Keep practicing, keep visualizing, and the geometry will reveal its secrets to you.