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

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

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

Step 1: Identify the type of ellipse

The given foci are \((\pm 2,0)\). Since the foci are on the x-axis, this is an ellipse centered at the origin with the major axis along the x-axis.

Step 2: Determine the distance between the center and the foci

The distance from the center \((0,0)\) to each focus \((\pm 2,0)\) is 2. This distance is denoted by \(c\). So, \(c = 2\).

Step 3: Use the eccentricity formula to find the semi-major axis

The eccentricity \(e\) is given by \(\frac{c}{a}\), where \(a\) is the semi-major axis. Here, \(e = \frac{1}{2}\) and \(c = 2\). Thus, \(\frac{2}{a} = \frac{1}{2}\). Solving for \(a\), multiply both sides by \(a\) and 2, to get \(4 = a\).

Step 4: Calculate the semi-minor axis

For an ellipse, the relationship between the axes and the foci is \(b = \sqrt{a^2 - c^2}\), where \(b\) is the length of the semi-minor axis. Substitute \(a = 4\) and \(c = 2\): \(b = \sqrt{4^2 - 2^2} = \sqrt{16 - 4} = \sqrt{12}\).

Step 5: Write the equation of the ellipse

The standard form of an ellipse with a major axis along the x-axis is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Substitute \(a = 4\) and \(b = \sqrt{12}\) into the equation: \(\frac{x^2}{16} + \frac{y^2}{12} = 1\).

Step 6: Verify with the options given

Compare the derived equation \(\frac{x^2}{16} + \frac{y^2}{12} = 1\) with the given options. It matches option (B).

Key Concepts

Foci of Ellipse

Ellipses have two special points known as foci (plural of focus), which play a crucial role in their geometric definition. For a given point on the ellipse, the sum of the distances to the two foci is constant. This property identifies an ellipse and helps differentiate it from other conic sections like circles or hyperbolas.
In the problem you were given, the foci are located at \((\pm 2, 0)\). This means they both lie on the x-axis, indicating the major axis of the ellipse is horizontal.

• The major axis passes through both foci and the center of the ellipse.
• If the foci are positioned symmetrically around the origin, this often defines the ellipse's orientation and center.

Understanding the placement of the foci is key to correctly identifying other ellipse parameters, like the semi-major and semi-minor axes.

Eccentricity

Eccentricity measures how "stretched" an ellipse is compared to a circle. It is a dimensionless number, denoted by \(e\), and ranges from 0 to 1 for ellipses. An eccentricity of 0 represents a perfect circle, while values closer to 1 indicate a more elongated shape.
In our exercise, the eccentricity is given as \(\frac{1}{2}\). This tells us that the ellipse is moderately oval, neither too circular nor extremely elongated.

• The eccentricity formula \(e = \frac{c}{a}\) relates the foci to the semi-major axis.
• Here, \(c = 2\) and solving with \(e = \frac{1}{2}\) provided \(a = 4\).

Remember:

• A very important insight is that the lower the eccentricity, the more the shape resembles a circle.

Semi-major and Semi-minor Axes

The semi-major and semi-minor axes are essential components in defining the dimensions of an ellipse.
The semi-major axis \(a\) is the longest diameter of the ellipse, stretching from one end to the other through the center and along the major axis. In standard form, the equation of an ellipse is given by \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), with \(a > b\).

• In this problem, the semi-major axis found was \(a = 4\).

The semi-minor axis \(b\) is the shortest diameter, perpendicular to the major axis through the center.

• Using the relation \(b = \sqrt{a^2 - c^2}\), you get \(b = \sqrt{12}\), indicating how the distance from the center to the edge of the ellipse in this direction is shorter.

These axes help graph the ellipse and determine its dimensions, making them crucial when solving or visualizing ellipse-related problems.