Problem 167 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

If the length of the latus rectum of an ellipse is 4 units and the distance between a focus and its nearest vertex on the major axis is \(\frac{3}{2}\) units, then its eccentricity is? (a) \(\frac{1}{2}\) (b) \(\frac{2}{3}\) (c) \(\frac{1}{9}\) (d) \(\frac{1}{3}\)

Step-by-Step Solution

Answer

The eccentricity is \(\frac{2}{3}\).

Step 1: Identify Given Values

We are provided with the length of the latus rectum of the ellipse, which is 4 units, and the distance between a focus and the nearest vertex on the major axis, which is \(\frac{3}{2}\) units.

Step 2: Understand the Ellipse Properties

The length of the latus rectum of an ellipse is given by the formula \( \frac{2b^2}{a} \), where \( a \) is the semi-major axis and \( b \) is the semi-minor axis of the ellipse. The distance from a focus to the nearest vertex is \( c \), where \( c = ae \) (\( e \) is the eccentricity).

Step 3: Relate Latus Rectum to Eccentricity

Using the formula for the latus rectum, we have \( \frac{2b^2}{a} = 4 \). Also, \( c = \frac{3}{2} \) is given, and since \( c = ae \), we can express \( e \) as \( e = \frac{c}{a} = \frac{\frac{3}{2}}{a} \).

Step 4: Use Relationships Among Parameters

We know the relationship between the semi-axes and the eccentricity: \( b^2 = a^2(1-e^2) \). We can also express \( b^2 \) from the latus rectum as \( b^2 = 2a \), because rearranging \( \frac{2b^2}{a} = 4 \) gives \( 2b^2 = 4a \).

Step 5: Solve for Eccentricity

Substitute \( b^2 = 2a \) into the equation \( b^2 = a^2(1-e^2) \), giving \( 2a = a^2(1-e^2) \). Simplify to \( 1-e^2 = \frac{2}{a} \), and knowing \( e = \frac{3}{2a} \), solve for \( e \) and find \( e^2 = \frac{4}{9} \). Then, use \( e^2 + (1-e^2) = 1 \).

Step 6: Complete Calculation

After substituting, solve the quadratic equation for \( e \) based on \( 1-e^2 = 1 - \frac{4}{9} = \frac{5}{9} \). Thus, \( e = \frac{2}{3} \).

Key Concepts

Latus Rectum

The latus rectum is a specific line segment associated with an ellipse that helps us understand more about its geometric properties. It is perpendicular to the major axis and passes through one of the foci. Intuitively, you can think of the latus rectum as the line that slices through the ellipse at its widest part through the focus.
The length of the latus rectum is determined by the equation \[ \text{Latus Rectum} = \frac{2b^2}{a} \]where \( a \) is the length of the semi-major axis and \( b \) is the length of the semi-minor axis. Understanding the latus rectum helps in calculating other key properties of an ellipse, such as its eccentricity.
In our problem, knowing that the length of the latus rectum is 4 units allows us to set up an equation based on the relationship provided. It serves as an excellent parameter to find the ellipse’s eccentricity.

Semi-Major Axis

The semi-major axis is one of the most crucial dimensions of an ellipse. Simply put, it is half of the longest diameter that spans across the ellipse, from one side to the other, through its center. This major axis has two important properties:

• It is the longest line that fits inside the ellipse, touching the curve at two opposing points.
• It is aligned with the x-axis in standard position for a horizontal ellipse.

The length of the semi-major axis plays a significant role in determining the ellipse's overall shape, eccentricity, and various distances crucial to solving problems related to ellipses. In our given problem, the semi-major axis was a key factor when calculating the eccentricity, providing the anchor for our formulas.

Semi-Minor Axis

The semi-minor axis is another fundamental aspect of an ellipse. It is the line segment that represents half of the shortest diameter across the ellipse, perpendicular to the semi-major axis.

• The semi-minor axis is perpendicular to the semi-major axis at the ellipse's center.
• It helps define the "height" or "width" of the ellipse depending on the orientation of the axes.

Knowing the semi-minor axis is essential as it is used in the formula for the latus rectum and influences the calculation of other ellipse properties, such as eccentricity and area. In our exercise, it helped us derive the relationship between the different axes and contributed to solving for the eccentricity.

Ellipse Properties

Ellipses have numerous interesting properties that distinguish them from other geometric shapes. Here are some key properties:

• The sum of the distances from any point on the ellipse to the two foci is constant.
• The eccentricity (\( e \)) measures how much an ellipse deviates from being a circle. It ranges from 0 (a perfect circle) to just below 1 (a very elongated shape).
• Two principal axes, the semi-major and semi-minor axes, define its dimensions.
• The formula for eccentricity given the axes is \( e = \sqrt{1 - \frac{b^2}{a^2}} \).

Understanding these properties provides a comprehensive view of how ellipses behave and allows one to solve for unknowns like eccentricity or semi-axes lengths using given data. In solving problems like ours, familiarity with these properties is essential to work through calculations effectively.