Problem 10 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the focal distance of an end of the minor axis of any ellipse (referred to its axes as the axes of \(x\) and \(y\) respectively) is \(k\) and the distance between the foci is \(2 h\), then its equation is (A) \(\frac{x^{2}}{k^{2}}+\frac{y^{2}}{k^{2}+h^{2}}=1\) (B) \(\frac{x^{2}}{k^{2}}+\frac{y^{2}}{h^{2}-k^{2}}=1\) (C) \(\frac{x^{2}}{k^{2}}+\frac{y^{2}}{k^{2}-h^{2}}=1\) (D) \(\frac{x^{2}}{k^{2}}+\frac{y^{2}}{h^{2}}=1\)

Step-by-Step Solution

Answer

The equation is incorrectly matched; none of the options are correct.

Step 1: Understand Ellipse Terminology

The focal distance from the center of an ellipse to a focus is denoted as \( c \). The distance from the center to any endpoint of the minor axis is \( b \), and the semi-major axis is \( a \). The relationship that defines an ellipse is \( c^2 = a^2 - b^2 \).

Step 2: Analyze Given Information

We are given the focal distance \( k \), which corresponds to \( \sqrt{a^2 - b^2} \), and \( 2h \) as the distance between the foci, meaning \( h = c \). Thus, \( c = h \).

Step 3: Set Up Equations Using Definitions

With the focal distance \( c = \sqrt{a^2 - b^2} = h \) and the formula \( c^2 = a^2 - b^2 \), we replace \( c \) with \( h \), resulting in the equation \( h^2 = a^2 - b^2 \). We must solve for \( b^2 \) in the context of the equation of an ellipse.

Step 4: Solve for \( b^2 \)

Rearrange the equation to solve for \( b^2 \), yielding \( b^2 = a^2 - h^2 \). Since \( b^2 = k^2 \) based on the given focal distance, we equate \( k^2 = a^2 - h^2 \).

Step 5: Derive Ellipse Equation

In the equation of an ellipse, we use \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Substitute \( b^2 = k^2 \) and \( a^2 = k^2 + h^2 \) into the equation to get \( \frac{x^2}{k^2 + h^2} + \frac{y^2}{k^2} = 1 \).

Step 6: Match Solution to Provided Options

Now we compare \( \frac{x^2}{k^2 + h^2} + \frac{y^2}{k^2} = 1 \) with the options. None of the options match directly, but this indicates an error.

Key Concepts

Focal Distance

In the world of ellipses, the focal distance is a key player. The focal distance in an ellipse refers to the distance from the center of the ellipse to one of its foci.
To imagine it better, consider the center of the ellipse as a midpoint. The points on the major axis that are two equal distances from this midpoint are the foci. These aren't located on the boundary of an ellipse, but rather inside it.
In a standard ellipse, there's a formula that describes the relationship between the focal distance and the axes dimensions: - The equation is represented as: \[ c^2 = a^2 - b^2 \] where: - \( c \) is the focal distance, - \( a \) is the semi-major axis length, - \( b \) is the semi-minor axis length. This formula is crucial, as it helps us understand how to derive or interpret the distances in any given ellipse.
Understanding this can be especially useful when solving for the lengths of various parts of an ellipse or establishing its equation.

Major and Minor Axes

A key aspect of understanding an ellipse is recognizing its two main axes: the major and minor axes. Both these axes play a central role in defining the ellipse's shape and size.
The major axis of an ellipse is the longest diameter that runs through the center from one side of the ellipse to the other. The minor axis, meanwhile, is the shortest.
The endpoints of these axes influence how elongated or stretched the ellipse appears. - To put this into perspective: - The semi-major axis is half the length of the major axis. - The semi-minor axis is half the length of the minor axis. To relate these to the standard ellipse equation:- The equation is: \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] where: - \( a \) corresponds to the semi-major axis, - \( b \) corresponds to the semi-minor axis. It's these axes that help you determine how much longer one side of the ellipse is compared to the other. In simpler terms, the greater the difference between \( a \) and \( b \), the more elongated the ellipse.

Distance Between Foci

Another significant property of an ellipse is the distance between its two foci. This distance greatly determines the shape of the ellipse.
If you recall, the foci are two points inside the ellipse. The major axis connects these foci and lies along its entire length.
Mathematically, this distance is expressed simply as \( 2c \), where \( c \) is the distance of each focus from the center of the ellipse. For this reason, we often say the distance between foci decides the 'flattening' of an ellipse. - Here's why this is important: - A circle is essentially a special type of ellipse where the distance between foci is 0, meaning both foci overlap at the center. - As this distance increases, the ellipse looks more stretched along the major axis. There is also a direct connection to the equation of an ellipse: - From the formula: \[ c = \sqrt{a^2 - b^2} \] we infer that knowing the lengths of the axes allows us to find the distance between foci.
Thus, when you appreciate the distance between the foci, you're one step closer to grasping the full picture of the ellipse's geometry.