An ellipse has its center at (1, -2), one focus at (3, -2) and one vertex at (5, -2). Then the lengt

Q: Wait, why is the major axis horizontal? How do I know it's not vertical?

Great question! Look at the coordinates provided: $C(1, -2)$, $F(3, -2)$, and $V(5, -2)$. Since the $y$-coordinate is $-2$ for all three points, they must lie on the horizontal line $y = -2$. If the ellipse were vertical, the $x$-coordinates would be the same, not the $y$-coordinates.

Q: Is there a difference between $b^2 = a^2 - c^2$ and $c^2 = a^2 - b^2$?

Mathematically, they are identical! It is just a matter of how you prefer to memorize it. Just remember that in an ellipse, $a$ is the largest value, so $a^2$ must be the sum of the other two squares. As long as you keep $a$ as the semi-major axis, you are on the right track.

Q: Why do we need $b^2$ to find the latus rectum?

The latus rectum is defined by the geometry of the ellipse's width at the focus. The formula $L = \frac{2b^2}{a}$ is derived directly from the ellipse equation $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$. Without $b^2$, we cannot define the 'thinness' or 'fatness' of the ellipse, which is essential for calculating the chord length at the focus.

Question

An ellipse has its center at $(1, -2)$, one focus at $(3, -2)$ and one vertex at $(5, -2)$. Then the length of its latus rectum is :

Options:

(A) 6

(B) $4\sqrt{3}$

(C) $\frac{16}{\sqrt{3}}$

(D) $6\sqrt{3}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing on a coordinate plane, looking at three points that define a beautiful, symmetrical curve: an ellipse. We have the center $C(1, -2)$, a focus $F(3, -2)$, and a vertex $V(5, -2)$.

Notice how all three points share the same $y$-coordinate of $-2$? This is not a coincidence; it is the key to the entire problem.

Because these points share a $y$-value, they lie on a horizontal line, which we identify as the major axis of our ellipse. This simple observation simplifies our entire path forward.

### Decoding the Parameters

In the world of ellipses, the distance from the center to the vertex is the semi-major axis, denoted by $a$. Since our center $C$ is at $(1, -2)$ and our vertex $V$ is at $(5, -2)$, the distance is simply the difference in their $x$-coordinates:
$$a = |5 - 1| = 4$$

Similarly, the focal distance $c$ is the distance from the center to the focus. With the center at $(1, -2)$ and the focus at $(3, -2)$, we calculate:
$$c = |3 - 1| = 2$$

We have now unlocked the two most important parameters of our ellipse: $a = 4$ and $c = 2$.

### The Golden Relation

Now, we reach the heart of the ellipse's identity. There is a fundamental relationship that binds the semi-major axis $a$, the semi-minor axis $b$, and the focal distance $c$ together:
$$b^2 = a^2 - c^2$$

This is the bridge that allows us to move from the major axis and focal distance to the width of the ellipse. Substituting our values, we get:
$$b^2 = (4)^2 - (2)^2$$

Calculating this, we find:
$$b^2 = 16 - 4 = 12$$

### The Final Calculation: The Latus Rectum

The question asks for the length of the latus rectum, which is the chord passing through the focus and perpendicular to the major axis. The formula for this length is:
$$L = \frac{2b^2}{a}$$

We have $b^2 = 12$ and $a = 4$. Plugging these into our formula, we get:
$$L = \frac{2(12)}{4}$$

This simplifies to:
$$L = \frac{24}{4} = 6$$

The elegance of this result is in its simplicity. We started with three points, identified the horizontal orientation, extracted the geometric parameters, and used the fundamental relations to find the latus rectum.

Always remember: in JEE problems, the geometry is your best friend. If you can visualize the points, the algebra will follow naturally. The final answer is **6**.

Animated Solution for Mathematics - Conic Sections: An ellipse has its center at $(1, - 2)$ , one focus at $(3, - 2)$ and one vertex at $(5, - 2)$ . Then the length of its latus rectum is :

Select Answer:

Visualized Solution

Visualize the Given Points

Identify the Major Axis

Define Semi-major Axis $a$

Compute $a$

Define Focal Distance $c$

Compute $c$

Fundamental Relation of an Ellipse

Substitute Values

Compute $b^{2}$

Visualize the Ellipse

Formula for Latus Rectum

Substitute into Latus Rectum Formula

Compute Final Length

Conclusion

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

Analyzing the Setup

Decoding the Parameters

The Golden Relation

The Final Calculation

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