If the x-intercept of a focal chord of the parabola y² = 8x + 4y + 4 is 3, then the length of this c

Q: Why do we need to complete the square? Can't we just use the original equation?

Completing the square is the key to unlocking the geometry of the parabola. The original equation is in a shifted form, which makes it difficult to identify the vertex and the focus directly. By transforming it into $(y - 2)^2 = 8(x + 1)$, we immediately see that it is a standard parabola $Y^2 = 4AX$ shifted to a new origin. This makes finding the focus and applying geometric properties much simpler.

Q: Is there a faster way to find the length of the focal chord than using the formula $L = 4A(1 + \frac{1}{m^2})$?

While you could find the intersection points by substituting the line equation into the parabola equation and then using the distance formula, that approach is algebraically intensive and prone to calculation errors. The formula $L = 4A(1 + \frac{1}{m^2})$ is a powerful tool derived from the properties of parabolas. In competitive exams like JEE, knowing and using such standard results saves precious time and reduces the chance of mistakes.

Q: What if the focal chord is vertical? Does the formula still work?

Great question! If the focal chord is vertical, its slope $m$ is undefined (or $\tan 90^\circ$). In this case, the formula $L = 4A(1 + \frac{1}{m^2})$ would involve division by zero. For a vertical focal chord (the latus rectum), the length is simply $4A$. The formula is specifically designed for non-vertical chords where the slope $m$ is defined.

Question

If the x-intercept of a focal chord of the parabola $y^2 = 8x + 4y + 4$ is 3, then the length of this chord is equal to

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, fellow traveler on the path to JEE mastery! Today, we are going to unravel the mystery of a focal chord. It might look like a daunting algebraic expression at first, but beneath the surface, it is a beautiful geometric dance.

Let us start by looking at the equation provided: $y^2 = 8x + 4y + 4$. This is a parabola, but it is hiding in a shifted coordinate system. To see its true nature, we must group the $y$ terms and complete the square.

By rearranging, we get $y^2 - 4y = 8x + 4$. Adding $4$ to both sides, we transform the left side into a perfect square: $(y - 2)^2 = 8x + 8$. Factoring out the $8$ on the right, we arrive at the elegant standard form:

$$(y - 2)^2 = 8(x + 1)$$

### Finding the Heart: The Focus

Now that we have the equation in the form $(y - k)^2 = 4A(x - h)$, where $(h, k)$ is the vertex and $A$ is the focal length, we can see that our vertex is at $(-1, 2)$. Comparing $4A$ with $8$, we find that $A = 2$.

This parameter $A$ is the heartbeat of the parabola; it tells us how 'wide' the curve is. In our shifted system, the focus is located at a distance $A$ from the vertex along the axis of symmetry.

Since our parabola opens to the right, the focus $S$ is at $(h + A, k)$, which is $(-1 + 2, 2)$, or simply $(1, 2)$. This point is the anchor for our focal chord.

### The Chord's Journey

A focal chord is a line segment that passes through the focus $S(1, 2)$. The problem gives us a crucial piece of information: the x-intercept of this chord is $3$. This means the line passes through the point $(3, 0)$.

Now, we have two points on our line: $(1, 2)$ and $(3, 0)$. Calculating the slope $m$ is straightforward:

$$m = \frac{0 - 2}{3 - 1} = \frac{-2}{2} = -1$$

We have successfully defined the line that cuts through our parabola.

### The Elegant Shortcut

We could find the intersection points of this line with the parabola and use the distance formula, but that is the long road. Instead, let us use the elegant property of focal chords.

The length $L$ of a focal chord with slope $m$ is given by the formula:

$$L = 4A \left(1 + \frac{1}{m^2}ight)$$

This formula is a gift to students, derived from the geometry of the parabola itself. Substituting our values, $A = 2$ and $m = -1$, we get:

$$L = 4(2) \left(1 + \frac{1}{(-1)^2}ight)$$

Simplifying this, we have $L = 8(1 + 1) = 8 	imes 2 = 16$. The length of the focal chord is **16 units**. It is truly satisfying when the math aligns so perfectly, isn't it?

Animated Solution for Mathematics - Conic Sections: If the x-intercept of a focal chord of the parabola $y^{2} = 8 x + 4 y + 4$ is 3, then the length of this chord is equal to

Enter Numerical Value:

Visualized Solution

Analyze the Parabola Equation

Complete the Square

Convert to Standard Form

Identify the Parameter $A$

Find the Coordinates of the Focus

Identify the Focal Chord

Calculate the Slope of the Chord

Formula for Length of Focal Chord

Substitute Values

Final Calculation

The Sigma Insight: Standard Equations of Parabola, Ellipse, and Hyperbola

Analyzing the Setup

Finding the Heart

The Chord's Journey

The Elegant Shortcut

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Animated Solution for Mathematics - Conic Sections: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the x-intercept of a focal chord of the parabola y2=8x+4y+4 is 3, then the length of this chord is equal to