Problem 81 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The equation of the ellipse whose foci are \((\pm 2,0)\) and eccentricity is \(\frac{1}{2}\) is: (A) \(\frac{x^{2}}{12}+\frac{y^{2}}{16}=1\) (B) \(\frac{x^{2}}{16}+\frac{y^{2}}{12}=1\) (C) \(\frac{x^{2}}{16}+\frac{y^{2}}{8}=1\) (D) none of these

Step-by-Step Solution

Answer

(B) \(\frac{x^{2}}{16} + \frac{y^{2}}{12} = 1\).

Step 1: Identify key information

First, identify the information given: the foci of the ellipse are \((\pm 2,0)\), and the eccentricity is \(\frac{1}{2}\). This indicates that the major axis is horizontal, centered at the origin, and the distance between the foci is 4.

Step 2: Relate eccentricity and foca

The formula for eccentricity is \( e = \frac{c}{a} \), where \(c\) is the distance from the center to a focus, and \(a\) is the semi-major axis length. Given the eccentricity \(e = \frac{1}{2}\) and \(c = 2\), we solve \(\frac{2}{a} = \frac{1}{2}\) to find \(a = 4\).

Step 3: Calculate semi-minor axis

With \(a = 4\) and using the relationship \(b^2 = a^2 - c^2\), calculate \(b\) as follows: \(b^2 = 4^2 - 2^2 = 16 - 4 = 12\). Hence, \(b = \sqrt{12}\).

Step 4: Form the equation of the ellipse

The standard form of an ellipse with a horizontal major axis is \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\). Substituting \(a = 4\) and \(b^2 = 12\), the equation becomes \(\frac{x^{2}}{16} + \frac{y^{2}}{12} = 1\).

Step 5: Check answer with options

Match the ellipse equation \(\frac{x^{2}}{16} + \frac{y^{2}}{12} = 1\) to the provided options. It corresponds to option (B).

Key Concepts

Eccentricity of Ellipse

The eccentricity of an ellipse is a measure of how much it deviates from being a perfect circle. It is represented by the symbol \( e \). In an ellipse, the eccentricity can vary between 0 and 1. When \( e = 0 \), the ellipse is a circle. Meanwhile, an eccentricity close to 1 means the ellipse is more stretched. For any ellipse, the formula for eccentricity is given by \( e = \frac{c}{a} \), where \( c \) is the distance from the center to each focus, and \( a \) is the length of the semi-major axis. In our exercise, the eccentricity is given as \( \frac{1}{2} \), which tells us it's moderately elongated.
Knowing the eccentricity helps in determining the shape of the ellipse and in distinguishing whether a given curve is indeed elliptical or not. The smaller the eccentricity, the closer the shape is to being circular. For practical problems, understanding eccentricity is essential for plotting and analyzing the ellipse accurately.

Semi-Major Axis

The semi-major axis of an ellipse is the longest radius stretching from the center to the edge of the ellipse, denoted by \( a \). It defines the maximum length of the ellipse and is crucial in determining its size. For a horizontal ellipse (as seen in our example), the semi-major axis runs along the x-axis. The center of the ellipse is equidistant between the foci, and the semi-major axis helps in calculating the overall dimensions of the ellipse.
The standard formula relating the semi-major axis \( a \) to other aspects of the ellipse is that \( a^2 = b^2 + c^2 \), where \( b \) is the semi-minor axis, and \( c \) is the distance to a focus. In our exercise, solving the equation \( \frac{2}{a} = \frac{1}{2} \) revealed that \( a = 4 \). This finding allows us to further investigate and confirm the equation of the ellipse itself.

Semi-Minor Axis

The semi-minor axis of an ellipse, denoted as \( b \), is the shortest radius from the center to the edge of the ellipse. It is perpendicular to the semi-major axis. While the semi-major axis determines the length, the semi-minor axis influences the curve's height or the squishiness of the ellipse.
In a horizontal ellipse, the semi-minor axis lies along the y-axis. To find \( b \), we can use the formula \( b^2 = a^2 - c^2 \). Here, \( a \) is the semi-major axis, and \( c \) is the focus distance. Calculations from the problem gave us \( b^2 = 12 \), leading to \( b = \sqrt{12} \). Recognizing how 'b' interacts with the ellipse's other components is necessary to understand its overall geometric properties.

Foci of Ellipse

Elliptical geometry is defined partially by its foci (plural of focus). An ellipse has two foci, located along its major axis. The unique property of an ellipse is that any point on its path keeps a constant total distance to the two foci. This relationship between foci and points on the ellipse helps in defining its shape.
For our given ellipse, the foci are \((\pm 2, 0)\), meaning they are positioned symmetrically around the center at the origin. The distance from the center to a focus \( c \) is crucial for calculating both the semi-major axis and the eccentricity: \( c = 2 \) in this exercise. Understanding the role of foci helps in visualizing and accurately constructing the shape of an ellipse.