Problem 90 - A Complete Resource Book in Mathematics for JEE Main

Question

An ellipse has \(O B\) as semi minor axis, \(F\) and \(F^{\prime}\) its focii and the angle \(F B F^{\prime}\) is a right angle. Then the eccentricity of the ellipse is (A) \(\frac{1}{\sqrt{2}}\) (B) \(\frac{1}{2}\) (C) \(\frac{1}{4}\) (D) \(\frac{1}{\sqrt{3}}\)

Step-by-Step Solution

Answer

The eccentricity of the ellipse is \( \frac{1}{\sqrt{3}} \) (Option D).

Step 1: Understand the Problem

We need to find the eccentricity of the ellipse, given that \( OB \) is the semi-minor axis and the angle \( FBF' \) is a right angle. The foci \( F \) and \( F' \) are symmetrically located along the major axis of the ellipse.

Step 2: Recall Ellipse Formulas

For an ellipse, the semi-major axis is \( a \), the semi-minor axis is \( b \), and the distance between the center of the ellipse and each focus \( c \) is given by the formula \( c = ae \), where \( e \) is the eccentricity. Also, \( a^2 = b^2 + c^2 \).

Step 3: Use the Right Angle Condition

The given condition is that \( \angle FBF' = 90^\circ \). Because \( O \), \( B \), \( F \), and \( F' \) form a right-angled triangle at \( B \), we use the Pythagorean theorem: \( OF^2 + OF'^2 = BF^2 \), where \( OF = c \) and \( BF = b \) because \( O \) is the center of the ellipse. Thus, \( b^2 = 2c^2 \).

Step 4: Substitute and Solve for Eccentricity

Substitute \( b^2 = 2c^2 \) into the formula \( a^2 = b^2 + c^2 \): \[ a^2 = 2c^2 + c^2 = 3c^2. \]Now, using \( c = a e \), we substitute \( a = \sqrt{3}c \) into the expression for \( \frac{c}{a} = e \): \[ e = \frac{c}{\sqrt{3}c} = \frac{1}{\sqrt{3}}. \]

Step 5: Verify the Solution

Re-examine the calculations to ensure there are no errors. The eccentricity \( e \) is found to be \( \frac{1}{\sqrt{3}} \), which matches option (D) given in the problem.

Key Concepts

Eccentricity

Every ellipse has an eccentricity, denoted by the letter "\(e\)." It helps to describe the shape of the ellipse. An eccentricity of 0 means the ellipse is a circle, while a value close to 1 indicates a very elongated shape.

To compute the eccentricity of an ellipse, you need values of the semi-major axis \(a\), the semi-minor axis \(b\), and the distance \(c\) from the center to any one of the focal points, known as the focus. The formula used is:

• Eccentricity formula: \(e = \frac{c}{a}\)

In our problem, the focal points, \(F\) and \(F'\), create a right angle with the semi-minor axis \(OB\). This condition gives us a clue that helps calculate the eccentricity. By following through the steps, we found \(e = \frac{1}{\sqrt{3}}\), linking the geometry of the ellipse to the formula.

Pythagorean Theorem

The Pythagorean theorem is a cornerstone of geometry. It relates the sides of a right-angled triangle. Easily remembered with the formula \(a^2 + b^2 = c^2\), it helps determine any side if the others are known.

In the context of the ellipse problem, it was crucial. We utilized it differently here due to the given condition of a right angle formed between the foci and a point on the semi-minor axis. With this information, the Pythagorean theorem helps us state:

• \(OF^2 + OF'^2 = BF^2\), where each focus is \(c\), and the semi-minor axis is \(b\).

Thanks to this application, we conclude that \(b^2 = 2c^2\), giving us an equation to calculate eccentricity using known ellipse properties.

Foci of Ellipse

The foci (singular: focus) are unique points located along the major axis of an ellipse. They hold a significant role in defining the ellipse's shape and properties.\To pinpoint the foci:

• The distance of each focus from the center is \(c\).
• The relationship involving the semi-major axis \(a\) is \(c = ae\), where \(e\) is the eccentricity.

In this problem, noticing the way focal points create a right angle with the ellipse's semi-minor axis allows us to apply geometric theorems effectively. The foci become central to understanding and solving the problem regarding the ellipse's eccentricity. Using this information, we find it consistent with proven mathematical concepts of ellipses.