Question

The ellipse \(E_{1}: \frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) is inscribed in a rectangle \(R\) whose sides are parallel to the coordinate axes. Another ellipse \(E_{2}\) passing through the point \((0,4)\) circumscribes the rectangle \(R\). The eccentricity of the ellipse \(E_{2}\) is (A) \(\frac{\sqrt{2}}{2}\) (B) \(\frac{\sqrt{3}}{2}\) (C) \(\frac{1}{2}\) (D) \(\frac{3}{4}\)

Short answer

\(\frac{\sqrt{3}}{2}\)

Step-by-step solution

Understanding ellipse E1 dimensions

The given ellipse E1 is defined by the equation \(\frac{x^{2}}{9} + \frac{y^{2}}{4} = 1\). The semi-major axis a1 is \(\sqrt{9} = 3\) and the semi-minor axis b1 is \(\sqrt{4} = 2\). Hence, the dimensions of the rectangle R are twice that of the semi-axes, giving us a width of 6 and a height of 4.

Identifying the dimensions of rectangle R

The lengths of the sides of rectangle R are parallel to the coordinate axes and equal to the lengths of the major and minor axes of E1. Therefore, the length and width of R are 6 units and 4 units respectively.

Determining the properties of ellipse E2

Ellipse E2 circumscribes rectangle R and passes through the point (0,4). The semi-minor axis b2 of E2 must be equal to half the height of R, which is 2 units. Since point (0,4) lies on E2, its semi-major axis a2 must be equal to 4 units, which is the y-coordinate of the point.

Calculating the eccentricity of ellipse E2

The eccentricity e of an ellipse is given by \(e = \sqrt{1 - \frac{b^{2}}{a^{2}}}\). Substituting the semi-major axis a2 = 4 and semi-minor axis b2 = 2, we get \(e = \sqrt{1 - \frac{2^{2}}{4^{2}}} = \sqrt{1 - \frac{1}{4}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}\).

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This method facilitates the link between algebra and geometry through the use of graphs of equations and curves. It allows for the precise calculation of lengths, areas, and other geometric properties by using algebraic equations.

For instance, in our exercise, the ellipse's equation \(E_{1}: \frac{x^{2}}{9}+\frac{y^{2}}{4}=1\) is a representation of an ellipse on the Cartesian plane. Through this equation, we observe the geometric place of points that satisfy the relationship, forming an ellipse centered at the origin \((0,0)\).

The properties of an ellipse can be understood within the context of a rectangle as shown by the descriptors of the ellipse's axes. The application of coordinate geometry makes it possible to analyze curves and shapes systematically for various applications, which is crucial for solving problems in JEE Advanced mathematics.

JEE Advanced Mathematics

JEE Advanced Mathematics encompasses a wide range of topics, including algebra, calculus, trigonometry, and geometry, each with its own set of applications in solving complex problems. The focus on understanding and applying concepts to solve problems is paramount.

In the context of the given problem, the concept of ellipse eccentricity combines elements of algebra and coordinate geometry. This underlines the integrative approach of JEE Advanced mathematics, where knowledge from multiple mathematical disciplines is used to approach a single problem.

A strong grasp of concepts such as the equation of an ellipse, eccentricity, and the relationship between an ellipse and a circumscribed rectangle requires a synthetic mastery of theoretical knowledge and problem-solving skills, which are heavily tested in the JEE Advanced exam.

Conic Sections

Conic sections are the curves obtained as the intersection of a plane with a cone. Depending on the angle of the cut, the resulting figure could be a circle, an ellipse, a parabola, or a hyperbola. Ellipses, the subject of our problem, are one type of conic section that occurs when the plane cuts the cone at an angle, such that it is parallel to one side of the cone but not to the base.

The general form of an ellipse in Cartesian coordinates is given by \(\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1\), where \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. The flexibility of conic sections in coordinate geometry, such as the ease of calculating the eccentricity, showcases why they are significant in the study of JEE Advanced mathematics. The eccentricity defines the 'ovalness' of an ellipse - a circle has an eccentricity of 0, and as the eccentricity approaches 1, the ellipse gets more elongated.

Moreover, understanding the relationship between the conic sections and geometric shapes such as rectangles, as in the case of the problem, is also critical. It shows the depth of intersection between geometric intuition and algebraic expression within the realm of advanced mathematics.