Problem 156 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the distance between the foci of an ellipse is 6 and the distance between its directrices is 12 , then the length of its latus rectum is: (a) \(\sqrt{3}\) (b) \(3 \sqrt{2}\) (c) \(\frac{3}{\sqrt{2}}\) (d) \(2 \sqrt{3}\)

Step-by-Step Solution

Answer

The length of the latus rectum is \(3\sqrt{2}\) (option b).

Step 1: Understand the Problem

The exercise gives us an ellipse with the distance between its foci as 6 and the distance between its directrices as 12. We are to find the length of its latus rectum from the given options.

Step 2: Basic Definitions

Recall the definitions: For an ellipse with semi-major axis as \(a\) and semi-minor axis as \(b\), the distance between foci is \(2c\), where \(c = \sqrt{a^2 - b^2}\). The distance between directrices is \(\frac{2a}{e}\), where \(e\) is the eccentricity of the ellipse and \(e = \frac{c}{a}\).

Step 3: Setup Equations

Given: \(2c = 6\) which implies \(c = 3\). The eccentricity \(e\) is given by \(e = \frac{c}{a} = \frac{3}{a}\). Also, the distance between directrices is \(\frac{2a}{e} = 12\).

Step 4: Solve for Semi-Major Axis

From the distance between directrices, \(\frac{2a}{e} = 12\) implies \(2a \times \frac{a}{3} = 12\) which simplifies to \(\frac{2a^2}{3} = 12\), leading to \(a^2 = 18\). Solving further, we find \(a = 3\sqrt{2}\).

Step 5: Find the Semi-Minor Axis

Using \(c = 3\), and \(a^2 = 18\), we have the equation for \(c\) as \(c = \sqrt{a^2 - b^2}\). So, \(3 = \sqrt{18 - b^2}\) gives \(b^2 = 9\). Hence, \(b = 3\).

Step 6: Calculate Latus Rectum Length

The length of the latus rectum \(L\) is \(\frac{2b^2}{a}\). Thus, substituting \(b = 3\) and \(a = 3\sqrt{2}\), we find \(L = \frac{2 \times 9}{3\sqrt{2}}= \frac{18}{3\sqrt{2}} = 3\sqrt{2}\).

Step 7: Determine the Correct Answer

The length of the latus rectum is \(3\sqrt{2}\). Thus, the correct answer from the options is (b).

Key Concepts

Latus Rectum

The concept of the latus rectum is crucial in understanding the properties of an ellipse. In simple words, the latus rectum is a line segment perpendicular to the major axis of the ellipse that passes through one of its foci. It is important because it gives insights into the size and shape of the ellipse at the foci.

In an ellipse, the length of the latus rectum,\[L = \frac{2b^2}{a}\]where \(a\) is the semi-major axis, and \(b\) is the semi-minor axis. It shows that this segment is influenced both by the width of the ellipse (through \(b\)) and by how stretched it is along the major axis (through \(a\)).

Understanding the latus rectum is useful as it relates to how the points are distributed around the foci. This has applications in fields where you need precise knowledge of an ellipse's properties, such as astronomy or physics.

Eccentricity

Eccentricity is a measure that helps us understand how much an ellipse deviates from being a perfect circle. For a circle, the eccentricity is zero, but for a more elongated ellipse, it approaches one.

The formula for eccentricity \(e\) of an ellipse is given by:\[e = \frac{c}{a}\]where \(c\) is the distance from the center to a focus, and \(a\) is the semi-major axis. This ratio captures the nature of the ellipse’s elongation. In our specific problem, it helped in forming the relationships needed to find the semi-major axis and eventually led to determining the correct length of the latus rectum.

In practice, knowing the eccentricity is invaluable in visualizing the orbit paths or the spread of elliptical objects. It indicates how stretched the ellipse is, providing critical insight into the ellipse’s geometry.

Semi-Major Axis

The semi-major axis is the longest radius of an ellipse and runs from the center through the farthest edge of the ellipse. It is denoted as \(a\) and is a key element in the intrinsic definition of the ellipse.

The semi-major axis, together with the semi-minor axis \(b\), defines the area and overall shape of the ellipse. The two are related through the equation:\[c = \sqrt{a^2 - b^2}\]Where \(c\) represents the distance from the center to one of the foci. This relationship highlights how the semi-major axis not only affects the size but also the positioning of the foci. Knowing \(a\) allows us to calculate fundamental properties like eccentricity and the latus rectum.

In this exercise, we solved for \(a\) using the provided distances, which was vital for determining other attributes of the ellipse. The semi-major axis is essential in delineating the geometry of the ellipse across many real-world applications.