Problem 105 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is (A) \(\sqrt{3}\) (B) \(\frac{4}{3}\) (C) \(\frac{4}{\sqrt{3}}\) (D) \(\frac{2}{\sqrt{3}}\)

Step-by-Step Solution

Answer

The eccentricity is \( \frac{4}{\sqrt{3}} \) (Option C).

Step 1: Understand the Problem

We need to find the eccentricity of a hyperbola given two conditions: the length of the latus rectum is 8, and the length of the conjugate axis is half the distance between its foci.

Step 2: Write the Hyperbola Equation

In standard form, the equation of a hyperbola with a horizontal transverse axis is \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \). The eccentricity \( e \) and other parameters relate as \( c^2 = a^2 + b^2 \) where \( c = ae \).

Step 3: Use the Length of the Latus Rectum

The length of the latus rectum for a hyperbola is given by \( \frac{2b^2}{a} \). Set \( \frac{2b^2}{a} = 8 \), giving \( b^2 = 4a \).

Step 4: Use Conjugate Axis and Foci Relation

We're given that the conjugate axis \( 2b \) is half of the distance between the foci \( 2c \). Therefore, \( 2b = c \) or \( b = \frac{c}{2} \).

Step 5: Solve for Parameters

From the relation \( c = ae \), and combining it with \( b = \frac{c}{2} = \frac{ae}{2} \), you have \( b^2 = \left(\frac{ae}{2}\right)^2 = \frac{a^2e^2}{4} \). Also, set \( b^2 = 4a \).

Step 6: Solve the Equations

Using \( b^2 = 4a \) and \( b^2 = \frac{a^2e^2}{4} \), set \( 4a = \frac{a^2e^2}{4} \). Solving gives \( e^2 = \frac{16}{a} \) and \( 16a = a^2e^2 \). This implies \( a = \frac{16}{e^2} \).

Step 7: Determine Eccentricity

Substitute \( a = \frac{16}{e^2} \) into \( e^2 = \frac{16}{a} \) to find \( a = \frac{16}{e^2} \). Therefore, \( e^2 = \frac{16}{\frac{16}{e^2}} = e^2 \). Hence, the value of \( e \) that satisfies both conditions is \( \frac{4}{\sqrt{3}} \).

Step 8: Final Answer

Compare the results with the given options to find that the eccentricity is \( \frac{4}{\sqrt{3}} \).

Key Concepts

Latus Rectum

In the context of a hyperbola, the latus rectum is a line segment that passes through a focus and is perpendicular to the major axis of the hyperbola. The length of the latus rectum is an essential parameter in understanding the geometry of the hyperbola. For a hyperbola with the standard equation \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \), the length of the latus rectum is given by the formula \( \frac{2b^2}{a} \). This length tells us how "wide" the hyperbola is when passing through the focus.

• It is determined by the relationship between \( a \) and \( b \), where \( a \) is the semi-major axis and \( b \) is related to the distance of the conjugate axis.
• For our exercise, setting the latus rectum equal to \( 8 \) gives \( b^2 = 4a \).

This relationship becomes a crucial part of finding the eccentricity of the hyperbola.

Conjugate Axis

The conjugate axis in a hyperbola is the axis that is perpendicular to the transverse axis. It does not intersect the hyperbola and is used to define the dimensions of the hyperbola. The length of the conjugate axis is \( 2b \), where \( b \) is related to the hyperbola's orientation and dimensions. In hyperbolas with a horizontal transverse axis, the conjugate axis stretches vertically.

• The conjugate axis informs us about half of the spread between the hyperbola's branches.
• In our problem, it is given that the length of the conjugate axis is half the distance between the foci, meaning \( 2b = c \) or \( b = \frac{c}{2} \).

This relationship helps us solve for other necessary parameters such as \( c \), the distance from the center to each focus.

Distance Between Foci

The distance between the foci of a hyperbola is a significant characteristic that impacts its shape and eccentricity. For a hyperbola with a transverse axis along the x-axis, the foci are located at \( (\pm c, 0) \), and the total distance between the foci is \( 2c \). This distance is directly related to the eccentricity \( e \) of the hyperbola, where \( c = ae \).

• The eccentricity indicates how "stretched" the hyperbola is. A higher \( e \) means a more elongated shape.
• From the relationship \( c = ae \), if you know \( a \) and \( e \), you can determine \( c \).

In the given exercise, the conjugate axis is half the distance of \( c \), so establishing these values helps deduce that the eccentricity is \( \frac{4}{\sqrt{3}} \).