Problem 91 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

In an ellipse, the distance between its foci is 6 and minor axis is 8 . Then its eccentricity is (A) \(\frac{3}{5}\) (B) \(\frac{1}{2}\) (C) \(\frac{4}{5}\) (D) \(\frac{1}{\sqrt{5}}\)

Step-by-Step Solution

Answer

The eccentricity is \( \frac{3}{5} \).

Step 1: Understand the problem

We are given an ellipse with certain properties: the distance between its foci is 6, and the length of the minor axis is 8. We need to find the eccentricity of the ellipse.

Step 2: Elucidate focal distance and relationship

The distance between the foci of the ellipse, 2c, is given as 6. Hence, the value of c is given by \( c = \frac{6}{2} = 3 \).

Step 3: Identify the relationship involving the semi-minor axis

The minor axis is given as 8, which means the semi-minor axis is \( b = \frac{8}{2} = 4 \).

Step 4: Formula for the ellipse relationship

The relationship between the semi-major axis \( a \), semi-minor axis \( b \), and focal distance \( c \) for an ellipse is \( c^2 = a^2 - b^2 \).

Step 5: Substitute known values into the formula

We know \( c = 3 \) and \( b = 4 \). Substitute these values into the formula: \( 3^2 = a^2 - 4^2 \). This simplifies to \( 9 = a^2 - 16 \).

Step 6: Solve for the semi-major axis

Rearrange and solve the equation for \( a^2 \): \( 9 + 16 = a^2 \), hence \( a^2 = 25 \), so \( a = 5 \).

Step 7: Calculate eccentricity

The eccentricity \( e \) of an ellipse is given by \( e = \frac{c}{a} \). Here, \( c = 3 \) and \( a = 5 \). So, \( e = \frac{3}{5} \).

Step 8: Verify the eccentricity

Double-check all calculated values: \( e = \frac{c}{a} = \frac{3}{5} \) matches the given choices, confirming our calculation.

Key Concepts

Eccentricity

The eccentricity of an ellipse is a measure that describes how much the ellipse deviates from being a perfect circle. A circle, for instance, has an eccentricity of 0 because it's perfectly round, while an ellipse has an eccentricity greater than 0 but less than 1. This is why ellipses can look stretched or elongated.

Eccentricity is mathematically defined as the ratio of the distance between the foci and the length of the major axis. In simple terms, it compares how far apart the foci are relative to the total length of the longest diameter of the ellipse. To compute the eccentricity (\( e \)), use the formula:

• \( e = \frac{c}{a} \)

where:

• \( c \) is the distance from the center of the ellipse to each focus, and
• \( a \) is the length of the semi-major axis.

In our example, we found that \( c = 3 \) and \( a = 5 \), resulting in an eccentricity of \( \frac{3}{5} \), indicating that the ellipse is not overly elongated.

Semi-major axis

The semi-major axis of an ellipse is the longest radius extending from the center to the edge of the ellipse. It's the primary indicator of the overall size of the ellipse, as it stretches across the major axis—the longest distance through the ellipse.

In mathematical terms, the semi-major axis (\( a \)) represents half of the full major axis length. If you know the semi-major axis, you can determine a lot about the shape and dimensions of the ellipse. It's used in the formula that connects it with the semi-minor axis and the focal distance:

• \( a^2 = b^2 + c^2 \)

where:

• \( b \) is the semi-minor axis, and
• \( c \) is the focal distance.

In our problem, we calculated \( a = 5 \) because the relationship between the semi-major axis, semi-minor axis (4), and focal distance (3) confirmed this measurement. Understanding the semi-major axis is essential for figuring out other attributes of the ellipse.

Foci

The foci (plural of focus) are two fixed points on the interior of an ellipse. Together, they help define the shape of the ellipse. The fundamental property of an ellipse is that the sum of the distances from any point on the ellipse to the two foci is constant.

This property is essential because it dictates how the ellipse elongates or stretches. The closer the foci are to each other, the more the ellipse resembles a circle. Conversely, as the foci move apart, the ellipse becomes more elongated. The distance between the foci (often denoted by \( 2c \)) is crucial because it helps determine the eccentricity.

In our specific example, the distance between the foci is 6, leading us to determine that \( c = 3 \) when divided by 2. The position of these foci in relation to the semi-major and semi-minor axes tells us about the stretch and direction of the ellipse. Understanding the foci allows us to grasp the basic nature of the ellipse shape, influencing other calculations like the eccentricity.