Focal Chord and Focal Properties

Focal Chord and Focal Properties: Your JEE Main Parabola Power-Up!

Hey JEE aspirants! Get ready to dive deep into the focal chord and its fascinating properties within the world of parabolas. This isn't just about memorizing formulas; it's about understanding the geometry and leveraging that understanding to solve problems quickly and accurately. Mastering these concepts is crucial for acing coordinate geometry questions in JEE Main. Let's unlock the secrets!

What is a Focal Chord?

Imagine a line passing through the focus of a parabola. If that line intersects the parabola at two distinct points, then that line segment joining those two points is called a focal chord. Simple as that! The focus is the heart of the parabola, and the focal chord gives us some special relationships.

Endpoints of a Focal Chord: The t₁t₂ = -1 Connection

Let's consider the standard parabola equation: $y^2 = 4ax$. Any point on this parabola can be represented as $(at^2, 2at)$, where $t$ is a parameter. Now, let's say a focal chord intersects the parabola at points $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$. Here's the magic:

Geometric Intuition: This relationship tells us that if you know the parameter of one endpoint of a focal chord, you automatically know the parameter of the other endpoint! It's a direct link. Think of it as a symmetry property around the focus.

Why is $t_1 t_2 = -1$? The slope of the line joining $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$ is $\frac{2a(t_2 - t_1)}{a(t_2^2 - t_1^2)} = \frac{2}{t_1 + t_2}$. For the line to be a focal chord, it has to pass through the focus $(a, 0)$. Applying the slope condition and solving yields $t_1t_2 = -1$.

Focal Distance: SP = |x + a| - Measuring from the Heart

The focal distance of a point $P(x, y)$ on the parabola is its distance from the focus $S(a, 0)$. It's denoted as $SP$. Here's a super useful formula:

Geometric Intuition: The focal distance is simply the distance of the point P from the directrix $x = -a$. In other words, it's just how far the point is horizontally from the directrix. This connects the focus and directrix in a fundamental way.

Why is $SP = |x + a|$? By definition of the parabola, the distance of any point on the parabola from the focus is equal to its distance from the directrix. The distance of a point $(x, y)$ from the directrix $x + a = 0$ is $|x + a|$.

Harmonic Mean Property of the Focal Chord: A Balancing Act

Let $P$ and $Q$ be the endpoints of a focal chord. The semi-latus rectum (half the length of the latus rectum) is the harmonic mean between the segments $SP$ and $SQ$. Remember, harmonic mean relates reciprocals.

Geometric Intuition: This property tells us that the lengths of the two focal chord segments, when combined in a specific reciprocal way, give us a constant value directly related to the parabola's parameter, $a$. It's a beautiful balance.

Why is $\frac{1}{SP} + \frac{1}{SQ} = \frac{1}{a}$? Let $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$ be the endpoints. Then $SP = a(1 + t_1^2)$ and $SQ = a(1 + t_2^2)$. Using $t_1t_2 = -1$, you can show that $\frac{1}{SP} + \frac{1}{SQ} = \frac{1}{a}$.

Length of the Focal Chord: Unleash the Power of the Parameter

We can express the length of the focal chord PQ in terms of the parameter $t_1$ (or $t_2$).

Since $t_1t_2 = -1$, we can write $t_2 = -\frac{1}{t_1}$. For simplicity, let $t_1 = t$, which makes $t_2 = -\frac{1}{t}$. Substituting into the formula $a(t_1 - t_2)^2$ gives us $a\left(t + \frac{1}{t}\right)^2$.

Length of Focal Chord at Angle θ:

Where $\theta$ is the angle the focal chord makes with the axis of the parabola.

Tips for Conquering Focal Chord Problems

Common Mistakes to Dodge

JEE-Specific Tricks

So there you have it! Focal chords and their properties, demystified and ready for your JEE Main assault. Practice these concepts, visualize the geometry, and you'll be solving parabola problems like a pro!