Problem 98 - A Complete Resource Book in Mathematics for JEE Main

Question

An ellipse is drawn by considering a diameter of the circle \((x-1)^{2}+y^{2}=1\) as its semi-minor axis and a diameter of the circle \(x^{2}+(y-2)^{2}=4\) as its semi-major axis. If the centre of the ellipse is the origin and its axes are the coordinate axes, then the equation of the ellipse is (A) \(4 x^{2}+y^{2}=4\) (B) \(x^{2}+4 y^{2}=8\) (C) \(4 x^{2}+y^{2}=8\) (D) \(x^{2}+4 y^{2}=16\)

Step-by-Step Solution

Answer

The equation of the ellipse is (A) \(4 x^{2}+y^{2}=4\).

Step 1: Identify the properties of the first circle

The equation of the first circle is \((x-1)^2 + y^2 = 1\). Comparing it with the standard form \((x-h)^2 + (y-k)^2 = r^2\), we identify that the center of the circle is \((1, 0)\) and the radius is 1. Hence, the diameter of this circle is 2.

Step 2: Find the semi-minor axis of the ellipse

The semi-minor axis of the ellipse is half of the diameter of the circle \((x-1)^2 + y^2 = 1\). Since the diameter is 2, the semi-minor axis is \(\frac{2}{2} = 1\).

Step 3: Identify the properties of the second circle

The equation of the second circle is \(x^2 + (y-2)^2 = 4\). From \((x-h)^2 + (y-k)^2 = r^2\), we find the center is \((0, 2)\) and the radius is 2. Thus, the diameter of this circle is 4.

Step 4: Find the semi-major axis of the ellipse

The semi-major axis of the ellipse is half of the diameter of the circle \(x^2 + (y-2)^2 = 4\). Since the diameter is 4, the semi-major axis is \(\frac{4}{2} = 2\).

Step 5: Formulate the equation of the ellipse

The ellipse is centered at the origin with its semi-major axis aligned with y-axis and semi-minor axis aligned with x-axis. Thus, the equation of the ellipse is \(\frac{x^2}{1^2} + \frac{y^2}{2^2} = 1\), simplifying to \(x^2 + \frac{y^2}{4} = 1\).

Step 6: Match the equation format to given options

Rewriting \(x^2 + \frac{y^2}{4} = 1\) as \(4x^2 + y^2 = 4\) by multiplying the entire equation by 4 gives us the equation that matches with option (A).

Key Concepts

semi-major axis

The semi-major axis is one of the most critical components of an ellipse. Imagine an ellipse as a stretched-out circle. The semi-major axis is the longest radius of the ellipse. It runs from the center to the furthest edge of the ellipse.
For the exercise, we see that the semi-major axis is aligned with the y-axis. It means the ellipse stretches more vertically.

• The semi-major axis of our ellipse is 2, derived from half of the diameter of the circle with equation \( x^2 + (y-2)^2 = 4 \).
• This circle's diameter is 4, leading us to its semi-major axis being \( \frac{4}{2} = 2 \).

In coordinate geometry, the semi-major axis is often denoted as \( a \). It is key to the ellipse's overall shape, affecting how elongated it appears. When solving problems involving ellipses, identifying the semi-major axis helps in writing the standard equation of the ellipse.

semi-minor axis

The semi-minor axis plays an equally important role in defining an ellipse's shape. It represents the shortest radius from the center to the edge, running perpendicular to the semi-major axis.For the ellipse in the given exercise, the semi-minor axis is oriented along the x-axis. This configuration means the ellipse is wider horizontally than vertically.

• The semi-minor axis measures 1, originating from the circle \((x-1)^2 + y^2 = 1\).
• Since the diameter of the circle is 2, the semi-minor axis becomes \( \frac{2}{2} = 1 \).

In coordinate geometry, the semi-minor axis is referred to as \( b \). It dictates the ellipse's compactness level along the axis perpendicular to the semi-major axis. The relative lengths of the semi-major and semi-minor axes give the ellipse its distinct shape.

coordinate geometry

Coordinate geometry, or analytic geometry, involves describing shapes using coordinate systems and algebraic equations. In this branch of mathematics, ellipses can be well-explained using geometric properties and algebraic equations. Understanding ellipses in coordinate geometry includes:

• Recognizing they are defined by two key axes: the semi-major and semi-minor axes. The lengths of these axes influence the equation of the ellipse.
• Locating the center, which in this exercise is the origin \((0, 0)\).
• Understanding that the standard form of an ellipse's equation in coordinate geometry is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) are the semi-major and semi-minor axes.

By applying coordinate geometry principles, we find the equation describing our ellipse as \( x^2 + \frac{y^2}{4} = 1 \). This approach elegantly bridges algebra with geometry, making it a powerful tool for solving problems related to ellipses and other shapes.