Problem 168 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The eccentricity of an ellipse having centre at the origin, axes along the co- ordinate axes and passing through the points \((4,-1)\) and \((-2,2)\) is: (a) \(\frac{1}{2}\) (b) \(\frac{2}{\sqrt{5}}\) (c) \(\frac{\sqrt{3}}{2}\) (d) \(\frac{\sqrt{3}}{4}\)

Step-by-Step Solution

Answer

The eccentricity of the ellipse is \( \frac{\sqrt{3}}{2} \), option (c).

Step 1: Establish the equation of the ellipse

Since the ellipse is centred at the origin and axes are along the coordinate axes, the equation of the ellipse can be represented in the standard form: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), where \(a\) and \(b\) are the semi-major and semi-minor axes respectively.

Step 2: Substitute point (4, -1) into the ellipse equation

Substitute \((x, y) = (4, -1)\) into the ellipse equation: \( \frac{4^2}{a^2} + \frac{(-1)^2}{b^2} = 1 \). This simplifies to: \( \frac{16}{a^2} + \frac{1}{b^2} = 1 \).

Step 3: Substitute point (-2, 2) into the ellipse equation

Substitute \((x, y) = (-2, 2)\) into the ellipse equation: \( \frac{(-2)^2}{a^2} + \frac{2^2}{b^2} = 1 \). This simplifies to: \( \frac{4}{a^2} + \frac{4}{b^2} = 1 \).

Step 4: Solve the system of equations

Now we have two equations: 1. \( \frac{16}{a^2} + \frac{1}{b^2} = 1 \) and 2. \( \frac{4}{a^2} + \frac{4}{b^2} = 1 \). Multiply equation 2 by 4: \( \frac{16}{a^2} + \frac{16}{b^2} = 4 \). Subtract equation 1 from this: \( \frac{15}{b^2} = 3 \). Solve for \( b^2 \): \( b^2 = 5 \). Now substitute \(b^2 = 5\) in equation 1 to find \(a^2\): \( \frac{16}{a^2} + \frac{1}{5} = 1 \), which simplifies to \( \frac{16}{a^2} = \frac{4}{5} \) and gives \( a^2 = 20 \).

Step 5: Calculate the eccentricity

The eccentricity of an ellipse is given by \( e = \sqrt{1 - \frac{b^2}{a^2}} \). Substituting the values \( a^2 = 20 \) and \( b^2 = 5 \) gives \( e = \sqrt{1 - \frac{5}{20}} = \sqrt{1 - \frac{1}{4}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \).

Key Concepts

Ellipse Equation

The equation of an ellipse is fundamental in coordinate geometry, especially when you need to study the properties of ellipses like eccentricity and the length of their axes. For an ellipse centered at the origin and with its axes aligned along the coordinate axes, the standard form of the ellipse equation is given by:

• \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)

In this equation, \(a\) represents the length of the semi-major axis, while \(b\) stands for the semi-minor axis. Together, they define the shape and size of the ellipse. Understanding this equation is the first step in exploring further properties such as eccentricity, which quantifies how stretched or elongated the ellipse is.

Semi-Major Axis

The semi-major axis is the longest radius of an ellipse. It stretches from the center of the ellipse to the furthest point on its perimeter. In mathematical terms, it is denoted by \(a\) in the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).

The value of \(a\) is crucial because it helps determine the ellipse's eccentricity. If the semi-major axis is equivalent to the semi-minor axis, the ellipse becomes a circle, which shows no elongation. Typically, when the problem provides particular points through which the ellipse passes, as we saw in our solution, this value is used to determine the length of both axes through solving systems of equations.

Semi-Minor Axis

The semi-minor axis represents the shortest radius from the center of the ellipse to its perimeter. This axis is shorter than the semi-major axis unless the ellipse is a perfect circle, where they are equal. It is denoted by \(b\) in our ellipse equation, \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \).

To find the value of \(b\), you often solve a system of equations that stem from substituting the ellipse's given points. Once \(a\) and \(b\) are determined, the ellipse’s eccentricity can be calculated using the formula \( e = \sqrt{1 - \frac{b^2}{a^2}} \), highlighting the importance of knowing both axes when exploring ellipses.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, provides a connection between algebra and geometry through graphs and equations. This field uses coordinates to represent geometric shapes, facilitating the study of their properties.

• In the case of an ellipse, coordinate geometry is used to express the curve in an equation form that can be analyzed algebraically.
• This approach allows for precise calculations of properties such as axes lengths and eccentricity, enhancing our understanding of geometric shapes in a systematic way.

The intersection of algebra and geometry via coordinates simplifies tasks like determining the ellipse's eccentricity, as coordinates help accurately describe its position and orientation in a plane.