Problem 159 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

An ellipse, with foci at \((0,2)\) and \((0,-2)\) and minor axis of length 4 , passes through which of the following points ? (a) \((\sqrt{2}, 2)\) (b) \((2, \sqrt{2})\) (c) \((2,2 \sqrt{2})\) (d) \((1,2 \sqrt{2})\)

Step-by-Step Solution

Answer

The point (b) \((2, \sqrt{2})\) lies on the ellipse.

Step 1: Understand the Ellipse Properties

The foci of the ellipse are at \((0, 2)\) and \((0, -2)\). This implies a horizontal major axis with center at \((0, 0)\). The distance between the foci (\(2c\)) is 4, which gives us \(c = 2\). The minor axis has length 4, making the semi-minor axis \(b = 2\).

Step 2: Determine the Semi-Major Axis

For an ellipse, the relationship between the semi-major axis \(a\), semi-minor axis \(b\), and the distance between foci \(c\) is \(c^2 = a^2 - b^2\). Here, \(b = 2\) and \(c = 2\), so substituting gives \(4 = a^2 - 4\) which simplifies to \(a^2 = 8\). Thus, \(a = \sqrt{8} = 2\sqrt{2}\).

Step 3: Write the Equation of the Ellipse

The standard equation for an ellipse with a horizontal major axis centered at the origin is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Substituting \(a^2 = 8\) and \(b^2 = 4\), the equation becomes \(\frac{x^2}{8} + \frac{y^2}{4} = 1\).

Step 4: Test Each Point in the Ellipse Equation

Substitute each point into the ellipse equation to check if it satisfies the equation:- **Point (a) \((\sqrt{2}, 2)\)**: \(\frac{(\sqrt{2})^2}{8} + \frac{2^2}{4} = \frac{2}{8} + \frac{4}{4} = \frac{1}{4} + 1 = \frac{5}{4}\), not 1.- **Point (b) \((2, \sqrt{2})\)**: \(\frac{2^2}{8} + \frac{(\sqrt{2})^2}{4} = \frac{4}{8} + \frac{2}{4} = \frac{1}{2} + \frac{1}{2} = 1\), correct.- **Point (c) \((2, 2\sqrt{2})\)**: \(\frac{2^2}{8} + \frac{(2\sqrt{2})^2}{4} = \frac{4}{8} + \frac{8}{4} = \frac{1}{2} + 2 = 2.5\), not 1.- **Point (d) \((1, 2\sqrt{2})\)**: \(\frac{1^2}{8} + \frac{(2\sqrt{2})^2}{4} = \frac{1}{8} + \frac{8}{4} = \frac{1}{8} + 2 = 2.125\), not 1.

Key Concepts

Ellipse Equation

In mathematics, the equation of an ellipse is a fundamental concept. An ellipse is a curved shape that kind of looks like a stretched out circle. To find the equation of an ellipse, you need to know certain properties like its center, major and minor axes. Here is the formula for the equation of an ellipse centered at the origin:

\[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \]
- **a** represents the length of the semi-major axis.- **b** represents the length of the semi-minor axis.
When this equation is given, it helps us figure out if a certain point lies on the ellipse. The point should satisfy the equation for it to be part of the ellipse. This means if you place that point into the equation, the left side should equal 1.

Foci of Ellipse

Ellipses have unique points called foci (singular: focus). Imagine two pins placed on a board with a loop of string around them. If you pull the string tight and move a pencil around, you create the shape of an ellipse. The pins would be at the foci.

For an ellipse:

• The foci are always situated along the major axis.
• They are equidistant from the center of the ellipse.

The distance between the center of the ellipse and each focus is represented by **c**. This parameter helps define the shape of the ellipse: - A larger value of **c** makes the ellipse more elongated. - If **c** equals zero, the ellipse becomes a perfect circle because the foci merge into one point at the center.
Understanding the foci is important because they help to calculate other dimensions of the ellipse, like the major and minor axes.

Semi-Major and Semi-Minor Axes

An ellipse has two main 'stretch' directions, characterized by its axes. These axes give the ellipse its shape:

• The **semi-major axis** is the longest radius of the ellipse. It runs from the center to the furthest edge along the longest direction. It's represented by **a** in the ellipse equation.
• The **semi-minor axis** is the shortest radius, running from the center to the closest edge along the shortest direction. It’s represented by **b**.

Together, the lengths of these axes define how 'stretched' or 'squished' the ellipse is. The semi-major and semi-minor axes are always perpendicular to each other, ensuring the curve is symmetrical. Knowing the values of **a** and **b** is crucial for sketching the ellipse and solving related mathematical problems, as they feature prominently in the ellipse equation. Hence, the equation and axes illustrate the harmonious structure of this geometric shape.