Problem 93 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

A focus of an ellipse is at the origin. The directrix is the line \(x=4\) and the eccentricity is \(\frac{1}{2} .\) Then the length of the semi-major axis is (A) \(\frac{8}{3}\) (B) \(\frac{2}{3}\) (C) \(\frac{4}{3}\) (D) \(\frac{5}{3}\)

Step-by-Step Solution

Answer

The length of the semi-major axis is \(\frac{8}{3}\).

Step 1: Understanding the Problem

The focus of the ellipse is at the origin \((0,0)\), and the directrix is the line \(x = 4\). The eccentricity \(e\) is given as \(\frac{1}{2}\). We need to find the length of the semi-major axis.

Step 2: Using Eccentricity Formula

The eccentricity \(e\) of an ellipse is given by the formula \(e = \frac{c}{a}\), where \(c\) is the distance from the center to the focus, and \(a\) is the semi-major axis.Given that \(e = \frac{1}{2}\), \(c = 0\) (since the focus is at the origin), and \(a\), we set up the relationship:\[\frac{1}{2} = \frac{4}{d + c} \] where \(d = 4\) (distance to directrix). Thus, \(c = a \cdot e\), so \(c = \frac{a}{2}\).

Step 3: Solving for Semi-Major Axis

We use the relationship derived earlier and substitute the known value:Substitute \(c = \frac{a}{2}\) back into our rearranged equation:\[ 4 = \frac{a}{2} + a \]Solving for \(a\):\(4 = \frac{3a}{2}\)\(a = \frac{8}{3}\).

Step 4: Final Calculation and Selection

Compare our final result to the options provided: (A) \(\frac{8}{3}\) (B) \(\frac{2}{3}\) (C) \(\frac{4}{3}\) (D) \(\frac{5}{3}\).Thus, the length of the semi-major axis is \(\frac{8}{3}\).

Key Concepts

Eccentricity

Eccentricity is a crucial concept when studying ellipses as it describes how elongated the shape is. It is defined by the letter 'e' and is calculated using the formula: \[ e = \frac{c}{a} \] where:

• \( e \) is the eccentricity,
• \( c \) represents the distance from the center of the ellipse to one of its foci, and
• \( a \) is the length of the semi-major axis.

The value of eccentricity gives us insight into the shape of the ellipse:

• An eccentricity of 0 indicates a perfect circle.
• For an ellipse, the eccentricity will be greater than 0 but less than 1.

In the given problem, an eccentricity of \( \frac{1}{2} \) shows that the ellipse is not very elongated. It is close to an oval shape, showing a moderate amount of stretching.

Semi-major axis

The semi-major axis is one of the most important measurements of an ellipse. It symbolizes half of the longest diameter of the ellipse. Represented by \( a \), it provides a foundation to understand the size and dimensions of the ellipse.In mathematics and geometry:

• The semi-major axis extends from the center of the ellipse to one end of the longest diameter.
• It is always greater than or equal to the semi-minor axis, which is half of the shortest diameter of the ellipse.

For the given exercise, the semi-major axis is calculated through the derived relationship between the given eccentricity and the directrix placement.By using the formula and solving the exercise, we found that the length of the semi-major axis is \( \frac{8}{3} \).Knowing the semi-major axis helps in better understanding and plotting the dimensions of an ellipse.

Directrix

The directrix is a fixed line used in the definition of an ellipse. It is not part of the ellipse but plays a crucial role in its formation and properties.Here's why the directrix matters:

• It is used along with a focus to define the ellipse in a plane.
• The ratio of the distance from any point on the ellipse to a focus, and from that point to the directrix, is always constant and equal to the eccentricity \( e \).

In our example, the directrix lies on the line \( x = 4 \). This line serves as a reference to establish the precise nature of the ellipse.Understanding the position of the directrix helps determine the relationship between the ellipse and its foci, which in turn, affects the calculation of both the eccentricity and the semi-major axis.