Problem 95 - A Complete Resource Book in Mathematics for JEE Main 2017

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

(A) \(x^2 + 16y^2 = 16\).

Step 1: Understand the given ellipse and rectangle

We have the ellipse \(x^2 + 4y^2 = 4\). This ellipse is inscribed in a rectangle which is aligned with the coordinate axes. An inscribed rectangle would have opposite sides touching the ellipse at its widest and tallest points. Thus, the semi-major axis along the x-axis is \(2\) and the semi-minor axis along the y-axis is \(1\). The dimension of the rectangle is \(4 \times 2\).

Step 2: Relate properties to outer ellipse

The rectangle inscribed in the first ellipse is also inscribed in another ellipse that goes through the point \((4,0)\). The inner rectangle's vertices on the clockwise from the positive x-axis are \((2,1), (-2,1), (-2,-1), (2,-1)\).

Step 3: Determine the outer ellipse's dimensions

The rectangle, with dimensions \(4\) by \(2\), must be inscribed within the second ellipse. Since this ellipse has to pass through \((4, 0)\), we know that \(a = 4\). Therefore, we also need that the semi-minor axis \(b\) causes the rectangle's height to fit. Since the semi-minor axis must exceed \(1\), we will determine \(b\).

Step 4: Formulate the outer ellipse equation

The outer ellipse has the equation \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Given that it passes through \((4,0)\), when \(x = 4\), \(\frac{16}{a^2} = 1\), we find \(a^2 = 16\), so \(a = 4\). Set \(\frac{16}{a^2} = 1\) and \(\frac{1}{b^2} = 1\), solving for \(b^2\).

Step 5: Solve for coefficient relations and verify

We need to verify possible relations using each option by substituting back into derived formulas. Testing multiple coefficients possible for the minor axis that reasonably fulfills the constraint must fit the maximum bounds of the rectangle \(2b = 2\) along minor's semi-axis \(b = 1\).Plug \(y = 1\) into valid dimensions gives the condition \(\frac{4y^2}{b^2}\); now retry, using setup implies \(16\) stretch rather than \(48\) when testing later.

Key Concepts

Ellipse Equation

An ellipse is a fascinating shape in geometry defined by a specific mathematical formula. The standard form of an ellipse equation is given by \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Here, \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively.

Understanding the equation involves recognizing how these axes determine the dimensions of the ellipse. In the exercise given, the inner ellipse equation \(x^2 + 4y^2 = 4\) can be rewritten in standard form by dividing all terms by 4, resulting in \(\frac{x^2}{4} + \frac{y^2}{1} = 1\). This tells us that the semi-major axis length \(a\) is 2, and the semi-minor axis length \(b\) is 1.

When dealing with problems involving ellipses, knowing how to manipulate the equation is crucial. For instance, rewriting it helps in identifying the lengths of the axes, helping solve geometric properties of the figure, whether inscribing it in a shape like a rectangle or locating it within a coordinate system.

Coordinate Geometry

Coordinate geometry is a branch of mathematics where algebraic equations represent geometrical figures. It's essential for visualizing entities like ellipses through equations.

In this context, an ellipse inscribed in a rectangle aligned with coordinate axes has its equation derived from positioning and dimensions. The exercise shows how to interpret coordinate points. Vertices at \((2,1), (-2,1), (-2,-1), (2,-1)\) are determined relative to the axes, reflecting an understanding of how a rectangle or ellipse is oriented in the plane.

• The rectangle's vertices are significant because they touch the ellipse exactly.
• The ellipse's properties affect how it's inscribed or fits within another ellipse or rectangle.

By using coordinate geometry principles, you can solve complex placement and dimension problems. It bridges algebraic operations with plane geometry, providing a clear path from equations to real-world shapes.

Major and Minor Axes

The major and minor axes of an ellipse are fundamental components that define its shape. The major axis is the longest diameter, and the minor axis is the shortest. These axes intersect at the ellipse's center.

In the exercise, understanding these axes is key. The semi-major axis \(a\) in the first ellipse \(x^2 + 4y^2 = 4\) is 2 along the \(x\)-axis, while the semi-minor axis \(b\) is 1 along the \(y\)-axis. This informs the dimensions of the rectangle inscribed inside the ellipse.

• Major Axis: This is the primary axis and runs horizontally in this case.
• Minor Axis: This is the secondary axis, running vertically.

Understanding the relationship and measurements of these axes is vital. It not only enables accurate calculations but also helps determine the full placement and orientation of the ellipse with respect to a surrounding shape, like the enclosing ellipse that passes through the point \((4,0)\). By knowing these axes, you can derive related geometry like the outer ellipse encompassing the rectangle.