Problem 95 - A Complete Resource Book in Mathematics for JEE Main

Question

The ellipse \(x^{2}+4 y^{2}=4\) is inscribed in a rectangle aligned with the coordinate axes, which in turn in inscribed in another ellipse that passes through the point \((4,0)\). Then the equation of the ellipse is (A) \(x^{2}+16 y^{2}=16\) (B) \(x^{2}+12 y^{2}=16\) (C) \(4 x^{2}+48 y^{2}=48\) (D) \(4 x^{2}+64 y^{2}=48\)

Step-by-Step Solution

Answer

The equation of the ellipse is Option (B), \(x^{2}+12y^{2}=16\).

Step 1: Identify the Characteristics of the First Ellipse

The equation of the first ellipse is \(x^{2} + 4y^{2} = 4\). This ellipse has its semi-major and semi-minor axes aligned with the coordinate axes. To rewrite this in standard form, divide everything by 4 to get \(\frac{x^2}{4} + \frac{y^2}{1} = 1\). From this, we identify the semi-major axis as 2 (along the x-axis) and the semi-minor axis as 1 (along the y-axis).

Step 2: Determine the Dimensions of the Inscribed Rectangle

The rectangle inscribed in the first ellipse will have its diagonally opposite corners touch the ellipse at points \( (\pm 2, 0) \text{ and } (0, \pm 1) \) based on the lengths of the semi-axes. Therefore, the length of the rectangle along the x-axis is 4, and along the y-axis is 2.

Step 3: Determine the Deputy Axes for the Second Ellipse

The rectangle is inscribed within a second ellipse that itself passes through the point (4, 0). The simplest ellipse equation format is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), given that the semi-major and semi-minor axes are the half-length of the side of the inscribed rectangle.

Step 4: Solve for the Second Ellipse Using Known Point

Since the rectangle's width is 4 and height is 2, the semi-major and semi-minor axes for the unknown ellipse are \(a = 4\) and \(b = 1\). Plug the point (4, 0) into \(\frac{x^2}{4^2} + \frac{y^2}{b^2} = 1\) which yields \(\frac{4^2}{4^2} = 1\), this is true, so \(a = 4\) is confirmed, meaning \(a^2 = 16\).

Step 5: Evaluate and Adjust Based on Option Format

We know the found ellipse happens to touch (4,0), therefore finding its form is based on ranges of formatting against provided selections. Based on the provided point it should match this equation: \(\frac{x^2}{4^2} + \frac{y^2}{b^2} = 1\) resulting in matching either a non-re-scaled version of head points graphically mapping to standardized ellipse forms.

Key Concepts

Ellipse Standard Form

When we talk about the standard form of an ellipse, we're referring to a specific way of writing the equation that helps us quickly understand its properties. The general standard form of an ellipse centered at the origin is \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here, \(a\) and \(b\) represent the semi-major and semi-minor axes, respectively.

In this form:

• \(a^2\) is always the square of the length of the semi-major axis.
• \(b^2\) is the square of the length of the semi-minor axis.

The semi-major axis is the longer diameter, while the semi-minor axis is the shorter. If \(a > b\), the ellipse is elongated along the x-axis. In our exercise, the first ellipse \(x^2 + 4y^2 = 4 \) becomes \(\frac{x^2}{4} + \frac{y^2}{1} = 1\), revealing \(a = 2\) and \(b = 1\).

This initial form is key to solving problems involving properties and positions of ellipses in relation to other shapes or points.

Inscribed Shapes

An inscribed shape is one that's drawn inside another such that all vertices of the inscribed shape lie on the boundary of the enclosing shape. In our context, we have a rectangle inscribed within the ellipse \(\frac{x^2}{4} + \frac{y^2}{1} = 1\).

For the rectangle to be perfectly inscribed:

• The corners of the rectangle must touch the ellipse at specific points.
• The sides of the rectangle are parallel to the axes.

For this ellipse, the rectangle's longest span (width of 4) aligns with the semi-major axis (x-direction), and its height of 2 aligns with the semi-minor axis (y-direction). This determines the span of the rectangle from \(x = -2\) to \(x = 2\) and from \(y = -1\) to \(y = 1\).

Understanding inscribed shapes is essential when setting up geometric relationships, such as in determining the dimensions of another ellipse within which this rectangle can be inscribed.

Semi-Major and Semi-Minor Axes

The semi-major and semi-minor axes are the foundation of understanding ellipses. They define the dimensions of the ellipse and describe its shape. The semi-major axis is the longest radius of the ellipse, stretching from the center to the farthest point on its boundary, while the semi-minor is the shortest.

In the given exercise, for the initial ellipse \(x^2 + 4y^2 = 4\), the axes dimensions show clarity when simplified:

• Semi-Major Axis: \(a = 2\).
• Semi-Minor Axis: \(b = 1\).

Every ellipse has its unique dimensions based on these axes, and they come into play again when determining the equation of the second larger ellipse that encompasses the rectangle and passes through a specified point, such as (4, 0). This larger ellipse leverages the rectangle's dimensions as it dictates \(a = 4\) and \(b = 1\), remaining consistent with the passage of the equation \(\frac{x^2}{16} + \frac{y^2}{1} = 1\).

Hence, comprehending these axes helps in translating geometric relationships into algebraic expressions and subsequently solving the problem.