If vertex of a parabola is (2, -1) and the equation of its directrix is 4x - 3y = 21, then the lengt

Q: Wait, why is the distance from the vertex to the directrix equal to $a$?

That is a fantastic question! By the geometric definition of a parabola, the distance from the focus to the vertex is $a$, and the distance from the vertex to the directrix is also $a$. This symmetry is a fundamental property that defines the shape of the parabola.

Q: Can I use the focus-directrix definition directly to solve this?

You certainly could, but it would be a much longer and more complex path. The vertex-directrix distance is a powerful shortcut derived from that very definition. In JEE, recognizing these geometric shortcuts is key to saving time.

Q: Is the latus rectum always $4a$?

Yes, absolutely! For any standard parabola, the length of the latus rectum is always $4a$, where $a$ is the distance from the vertex to the focus (or the vertex to the directrix). It is a constant, elegant relationship.

Question

If vertex of a parabola is $(2, -1)$ and the equation of its directrix is $4x - 3y = 21$, then the length of its latus rectum is

Options:

(A) 2

(B) 8

(C) 12

(D) 16

Sigma Percentile · Accepted Answer

### The Geometry of the Parabola: A Journey of Discovery

Imagine you are standing on the Cartesian plane, looking at a parabola. It is not just a curve; it is a locus of points, a perfect balance between a fixed point called the focus and a fixed line called the directrix.

Today, we are going to unravel the mystery of a specific parabola, given its vertex $V(2, -1)$ and its directrix $4x - 3y - 21 = 0$. Our goal is to find the length of its latus rectum.

This might seem like a daunting algebraic task, but if we look at it through the lens of geometry, it becomes a beautiful, logical progression.

### Phase 1: The Geometric Anchor

First, let us orient ourselves. We are given the vertex $V(2, -1)$ and the directrix $L: 4x - 3y - 21 = 0$.

The vertex is the point where the parabola turns, and the directrix is the line that defines its shape. There is a fundamental, elegant property here: the perpendicular distance from the vertex to the directrix is exactly equal to a constant we call $a$.

This constant $a$ is the master key. It dictates the 'width' of the parabola. If we find $a$, we find the heart of the parabola.

### Phase 2: The Bridge (The Distance Formula)

Now, how do we find this distance $a$? We have a point $V(2, -1)$ and a line $4x - 3y - 21 = 0$.

This is a classic application of the perpendicular distance formula from a point $(x_1, y_1)$ to a line $Ax + By + C = 0$, which is given by:

$$d = \frac{|Ax_1 + By_1 + C|}{\sqrt{A^2 + B^2}}$$

Let us plug in our values. Here, $x_1 = 2$, $y_1 = -1$, $A = 4$, $B = -3$, and $C = -21$. The calculation looks like this:

$$a = \frac{|4(2) - 3(-1) - 21|}{\sqrt{4^2 + (-3)^2}}$$

Take a breath. Let us simplify the numerator: $4(2) = 8$, $-3(-1) = 3$, so we have $|8 + 3 - 21| = |11 - 21| = |-10| = 10$.

Now for the denominator: $\sqrt{16 + 9} = \sqrt{25} = 5$. Thus, $a = \frac{10}{5} = 2$. We have found our master key; the distance $a$ is $2$.

### Phase 3: The Latus Rectum

Finally, we arrive at the question: what is the length of the latus rectum? The latus rectum is the chord passing through the focus, perpendicular to the axis of the parabola.

Its length is a standard result in conic sections: it is always $4a$. Since we have already determined that $a = 2$, the calculation is straightforward:

$$	ext{Length} = 4 	imes a = 4 	imes 2 = 8$$

And there we have it! The length of the latus rectum is **8 units**. It is not just about the number; it is about the journey from the geometric definition to the final, elegant result.

Animated Solution for Mathematics - Conic Sections: If vertex of a parabola is $(2, - 1)$ and the equation of its directrix is $4 x - 3 y = 21$ , then the length of its latus rectum is

Select Answer:

Visualized Solution

Visualizing the Given Data

The Geometric Connection

The Distance Formula

Substituting the Values

Simplifying the Numerator

Simplifying the Denominator

Finding the value of $a$

The Latus Rectum

Final Calculation

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

The Geometry of the Parabola

Phase 1

Phase 2

Phase 3

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