RD Sharma Class 11 Chapter 26 Solutions (Ellipse)
RD Sharma Solutions for Class 11 Maths Chapter 26 ‘Ellipse’ are geared towards providing you in-depth knowledge on ellipses which will be useful in competitive exams like IIT JEE or NEET. In this chapter, you would learn about the equation of an ellipse in its standard form as well as other forms. You would also gain knowledge on other topics around ellipses like tracing an ellipse, ordinate, latus-rectum, double ordinate, the focal distance of any point in an ellipse, second directrix of an ellipse, the second focus of an ellipse, vertices, directrices, major/minor axes, and foci.
This chapter has a total of 76 questions divided into 3 sections, which go over many exercises brushing up your knowledge on ellipses. Questions range from very short answers to objective-type MCQs to simple and complex questions on the equation of an ellipse. RD Sharma Class 11 Maths Solutions Chapter 26 will make you thorough with ellipses so that you can take up any competitive exam with confidence.
We provide exercise-wise solutions to this chapter with very clear and easy steps to follow, even for the problems which seem complex. Using the methods and tools you should be able to manage your time efficiently in exams.
Topics Covered in RD Sharma Solutions for Class 11 Maths Chapter 26 – Ellipse
- Ellipse fundamentals – An ellipse is a plane curve that is formed when a plane intersects a cone making an angle at the base. Below are the main characteristics of an ellipse:
- An ellipse has 2 focal points called foci which are 2 points on either side of the major axis at equal distance from the center.
- If you sum up the distance between the 2 foci and every point on an ellipse, it would be the same.
- Center of an ellipse is the middle point between its 2 foci.
- Vertices are the endpoints of a line which crosses its center. The line is the major axis.
- The minor axis is the line that is perpendicular to the major axis and passes through its center.
- The endpoints of a minor axis are called its co-vertices.
- If an ellipse has a major axis length = 2 * a, minor axis length = 2 * b and has foci coordinates as (0,±c) : c² = a² – b²
Standard form equation of an ellipse: For an ellipse which has center at (c1,c2), has its major axis parallel to the x-axis, and (p1, p2) is a point on the ellipse:
((p1 – c1)² / a²) + ((p2 – c2)² / b²) = 1
If its major axis is parallel to the y-axis then the equation would be
((p1 – c1)² / b²) + ((p2 – c2)² / a²) = 1
The eccentricity of an ellipse – The eccentricity of an ellipse defines how elongated the ellipse is. An eccentricity of 0 would mean a circle and an eccentricity of 1 means a fully elongated ellipse.
Eccentricity = c / a
c = distance from center of an ellipse to its focus
a = distance from center to a vertex
Latus Rectum of an ellipse – A line that passes through any of the foci and is perpendicular to the major axis of an ellipse and has endpoints on the ellipse is called its latus-rectum.
Length of latus rectum = (2 * b²) / a
a = ½ of the major axis
b = ½ minor axis
and a > b
Directrix of an ellipse – A parallel line drawn outside an ellipse, perpendicular to its major axis is called its directrix.
directrix = ±a/e Here a = ½ major axis e = eccentricity of the ellipse
The ordinate and Double ordinate of an ellipse – If p1 is a point on ellipse and perpendicular is drawn from p1 to the major axis of an ellipse which meets the ellipse’s major axis at m1, then p1m1 is an ordinate of the ellipse.
If the point of contact is extended to the ellipse periphery where it meets at m2 then p1m2 is its double ordinate.
Auxiliary Circle – Circle is a special form of ellipse where its major axis = minor axis. If one draws a circle inside an ellipse with the major axis as its diameter then that is an auxiliary circle and its equation is
p1² + p2² = a²
p1, p2 are any points on the ellipse and a is ½ major axis.
Parametric equation of an ellipse – The equation is:
p1 = a cos t
p2 = b sin t
Here p1,p2 are coordinates of any point on the ellipse
a = ½ major axis
b = ½ minor axis
t = a parameter with a range between 0 to 2π radians
Discussion of Exercises of RD Sharma Solutions for Class 11 Maths Chapter 26
- The first set of exercises has 43 questions that revolve around the equation of an ellipse. You will be given some parameters like foci, directrices and need to find the equation.
- The second set of exercises has 9 questions that have short answers. They brush up your knowledge on eccentricity, foci, and latus rectum of an ellipse
- The third set of exercises is MCQ or objective-type and has 24 questions. They cover all topics related to an ellipse.
Benefits of RD Sharma Solutions for Class 11 Maths Chapter 26 by Instasolv
Instasolv makes learning ellipse equations very simple with easy to follow steps and diagrams. Our RD Sharma Solutions for Class 11 Chapter 26 are designed by the best subject matter experts who have years of knowledge and experience. We have kept all the solutions as per the CBSE guidelines so that you can easily prepare for your school exams as well.