In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor a

Q: Wait, why is the major axis $2a$ and not $2b$ here? I always get confused with the vertical ellipse!

That is a classic trap! In an ellipse, $a$ is always defined as the semi-major axis, meaning it is the larger value. When the focus is on the y-axis, the ellipse is 'tall,' so the y-denominator must be the larger one ($a^2$). Always associate $a$ with the direction of the foci.

Q: Why can't we just use $e = \frac{c}{a}$? Do we really need the $a^2 - b^2 = c^2$ relation?

You absolutely can! Using $e = \frac{c}{a}$ is perfectly valid. However, in JEE problems, working directly with $c^2 = a^2 - b^2$ (where $c = ae$) often saves you from dealing with square roots of eccentricity, making your algebra much cleaner and less prone to calculation errors.

Q: How do I know if the latus rectum formula is $\frac{2b^2}{a}$ or $\frac{2a^2}{b}$?

Think of the latus rectum as the 'width' of the ellipse at the focus. It is always $2 \times \frac{(\text{semi-minor})^2}{\text{semi-major}}$. Since $a$ is always the semi-major axis, the formula is always $2 \times \frac{b^2}{a}$. It never changes, regardless of orientation!

Q: Is it possible for $a$ and $b$ to be equal? What happens to the latus rectum then?

If $a = b$, the ellipse collapses into a circle! In that case, $a^2 - b^2 = 0$, which means the focus distance $c = 0$. The foci merge at the center, and the latus rectum formula $\frac{2b^2}{a}$ simplifies to $2a$, which is the diameter of the circle.

Q: I got $a=10$ and $b=5$. Does the order matter for the final answer?

It matters immensely! If you swapped them, you would get a latus rectum of $\frac{2(100)}{5} = 40$, which is incorrect. Always verify that $a > b$ before calculating the final ratio. Your check is simple: $a$ must be the larger value.

Question

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at $(0, 5\sqrt{3})$, then the length of its latus rectum is:

Options:

(A) 10

(B) 8

(C) 5

(D) 6

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine you are standing at the origin of a coordinate plane. You are tasked with defining the shape of an ellipse, a beautiful, symmetric curve that governs everything from the orbits of planets to the acoustics of whispering galleries.

We are given two vital clues: the center is at the origin $(0, 0)$, and one of the foci sits proudly at $(0, 5\sqrt{3})$.

### Visualizing the Orientation

Before we touch a single algebraic symbol, let us visualize. The focus lies on the $y$-axis. In the language of geometry, this tells us immediately that our ellipse is 'vertical'—it is stretched along the $y$-axis like a tall, elegant vase.

Because the ellipse is vertical, the major axis lies along the $y$-axis. We define the semi-major axis as $a$ and the semi-minor axis as $b$. Consequently, our standard equation takes the form:

$$\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1$$

where $a > b$. This is our foundation. If we lose sight of this orientation, the entire problem will tilt, so keep this verticality in your mind's eye.

### Translating the Clues

The problem gives us a gift: the difference between the lengths of the major and minor axes is $10$. The major axis is $2a$ and the minor axis is $2b$. Thus, we write:

$$2a - 2b = 10 \implies a - b = 5$$

We shall call this **Equation 1**.

Now, consider the focus. The distance from the center to the focus is defined as $ae$. We are given this distance as $5\sqrt{3}$.

We know the fundamental identity for any ellipse is $a^2 - b^2 = (ae)^2$. Substituting our focus value, we get:

$$a^2 - b^2 = (5\sqrt{3})^2 = 25 	imes 3 = 75$$

This is **Equation 2**. Look at the beauty of this structure! We have a difference of squares.

### The Algebraic Symphony

This is where the magic happens. We have $a^2 - b^2 = 75$ and we know from algebra that $a^2 - b^2 = (a - b)(a + b)$. We already know $a - b = 5$ from our first clue.

Substituting this into our identity:

$$5(a + b) = 75$$

Dividing both sides by $5$, we find the sum of the semi-axes: $a + b = 15$. Now, we have a simple system of linear equations:

$$a - b = 5$$
$$a + b = 15$$

Adding these two equations yields $2a = 20$, so $a = 10$. Subtracting them yields $2b = 10$, so $b = 5$. We have successfully unmasked the dimensions of our ellipse.

### The Final Calculation

We are asked for the length of the latus rectum. The latus rectum is the chord passing through the focus, perpendicular to the major axis. For any ellipse, its length is given by the elegant formula:

$$LR = \frac{2b^2}{a}$$

Substituting our values $a = 10$ and $b = 5$:

$$LR = \frac{2(5^2)}{10} = \frac{2(25)}{10} = \frac{50}{10} = 5$$

And there it is. The length of the latus rectum is **5**.

### Reflection

Do you see how the geometry dictated the algebra? By identifying the orientation first, we avoided the trap of misassigning $a$ and $b$.

By using the difference of squares, we bypassed the need for tedious substitution. Physics and mathematics are not just about grinding through numbers; they are about finding the most elegant path through the forest. You have mastered the ellipse today—keep that confidence as you tackle the next challenge!

Animated Solution for Mathematics - Conic Sections: In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at $(0, 53)$ , then the length of its latus rectum is:

Select Answer:

Visualized Solution

Identifying the Focus and Orientation

Defining the Axes Lengths

Simplifying the Axes Equation

The Focus Relation

Substituting the Focus Value

Squaring the Focus Value

Factoring the Difference of Squares

Finding the Sum of Semi-Axes

Solving for $a$ and $b$

The Latus Rectum Formula

Substituting $a$ and $b$ into LR

Final Calculation

Conclusion

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

Analyzing the Setup

Visualizing the Orientation

Translating the Clues

The Algebraic Symphony

The Final Calculation

Reflection

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