Vedantu’s Class 11 Chapter 26 - Ellipse Ex 26.1 RD Sharma Solutions
FAQs on RD Sharma Class 11 Solutions Chapter 26 - Ellipse (Ex 26.1) Exercise 26.1
1. What is the standard equation of an ellipse as required in RD Sharma Class 11 solutions?
The standard equation of an ellipse centered at the origin (0,0) depends on the orientation of its major axis. The two forms you will use to solve problems in Exercise 26.1 are:
When the major axis is along the x-axis: x²/a² + y²/b² = 1
When the major axis is along the y-axis: x²/b² + y²/a² = 1
Here, 'a' represents the length of the semi-major axis and 'b' represents the length of the semi-minor axis, with the condition that a > b.
2. What key parameters are typically given in RD Sharma Chapter 26, Exercise 26.1 problems to find the ellipse's equation?
To solve the problems in this exercise, you will usually be provided with a combination of the following parameters:
Coordinates of the vertices: These points mark the ends of the major axis.
Coordinates of the foci: These are two fixed points inside the ellipse.
Length of the major and minor axes: Used to directly find the values of 'a' and 'b'.
Eccentricity (e): A ratio that defines the shape of the ellipse.
Length of the latus rectum: The chord passing through a focus, perpendicular to the major axis.
By using the relationships between these parameters, you can determine the values of 'a' and 'b' to form the final equation.
3. How can a student determine if the major axis of an ellipse is horizontal or vertical when solving a problem?
You can determine the orientation of the major axis by observing the given coordinates. This is a critical first step for choosing the correct standard equation:
If the coordinates of the foci or vertices have non-zero x-values and y-values of 0 (e.g., (±c, 0) or (±a, 0)), the major axis lies along the x-axis.
If the coordinates of the foci or vertices have x-values of 0 and non-zero y-values (e.g., (0, ±c) or (0, ±a)), the major axis lies along the y-axis.
4. What is the fundamental relationship between the semi-major axis (a), semi-minor axis (b), and the foci distance (c)?
The relationship between the semi-major axis (a), semi-minor axis (b), and the distance from the center to a focus (c) is crucial for almost every problem. The formula is c² = a² - b². This equation allows you to calculate one of the three values if the other two are known. For instance, if you are given the vertices (which gives 'a') and the foci (which gives 'c'), you can easily solve for 'b' to complete the ellipse's standard equation.
5. How does an ellipse differ from a circle in terms of its equation and key properties?
The primary difference lies in their axes and foci. A circle is a special case of an ellipse where the major and minor axes are equal (a = b). Consequently:
Equation: An ellipse has denominators a² and b², which are different. For a circle, a² = b², so the equation x²/a² + y²/a² = 1 simplifies to x² + y² = a².
Foci: An ellipse has two distinct foci. In a circle, both foci coincide at the center (c=0), because a=b in the relation c² = a² - b².
Eccentricity: An ellipse has an eccentricity between 0 and 1 (0 ≤ e < 1). A circle has an eccentricity of exactly 0.
6. What does the eccentricity of an ellipse represent, and why is its value always between 0 and 1?
The eccentricity, defined as e = c/a, measures how much an ellipse deviates from being a perfect circle. It describes the 'flatness' or 'ovalness' of the shape. Its value is always in the range 0 ≤ e < 1 because:
If e = 0, then c = 0. This means the foci are at the center, so a = b, and the shape is a circle.
As 'e' increases towards 1, the value of 'c' gets closer to 'a', making the ellipse more elongated and flatter.
The value 'c' (distance from center to focus) can never be greater than or equal to 'a' (distance from center to vertex) for an ellipse, which is why 'e' must be less than 1.
7. What is a common mistake to avoid when finding the equation of an ellipse for questions in Exercise 26.1?
A common mistake is incorrectly identifying the major axis and subsequently mixing up 'a²' and 'b²' in the standard equation. Always remember that 'a' is the semi-major axis, so a² is always the larger denominator. If the vertices are on the y-axis, the major axis is vertical, and the larger denominator (a²) must be placed under the y² term. Rushing this first step often leads to an incorrect final equation.
