Question 1.
Find the
(i) lengths of the principal axes
(ii) co-ordinates of the foci
(iii) equations of directrices
(iv) length of the latus rectum
(v) distance between foci
(vi) distance between directrices of the ellipse:
a = 5 and b = 3
Since a > b,
X-axis is the major axis and Y-axis is the minor axis.
(i) Length of major axis = 2a = 2(5) = 10
Length of minor axis = 2b = 2(3) = 6
Lengths of the principal axes are 10 and 6.
![Maharashtra Board Class 11 Maths Part 1 Chapter 7 Conic Sections Ex 7.2 Solution 2 word image 18220 2](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18220-2.png)
![Maharashtra Board Class 11 Maths Part 1 Chapter 7 Conic Sections Ex 7.2 Solution 3 word image 18220 3](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18220-3.png)
![Maharashtra Board Class 11 Maths Part 1 Chapter 7 Conic Sections Ex 7.2 Solution 4 word image 18220 4](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18220-4.png)
(i) Length of major axis = 2a = 2(2) = 4
Length of minor axis = 2b = 2√3
Lengths of the principal axes are 4 and 2√3.
![Maharashtra Board Class 11 Maths Part 1 Chapter 7 Conic Sections Ex 7.2 Solution 5 word image 18220 5](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18220-5.png)
a = √3 and b = 1
Since a > b,
X-axis is the major axis and Y-axis is the minor axis.
(i) Length of major axis = 2a = 2√3
Length of minor axis = 2b = 2(1) = 2
Lengths of the principal axes are 2√3 and 2.
![Maharashtra Board Class 11 Maths Part 1 Chapter 7 Conic Sections Ex 7.2 Solution 7 word image 18220 7](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18220-7.png)
![Maharashtra Board Class 11 Maths Part 1 Chapter 7 Conic Sections Ex 7.2 Solution 8 word image 18220 8](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18220-8.png)
![Maharashtra Board Class 11 Maths Part 1 Chapter 7 Conic Sections Ex 7.2 Solution 9 word image 18220 9](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18220-9.png)
![Maharashtra Board Class 11 Maths Part 1 Chapter 7 Conic Sections Ex 7.2 Solution 10 word image 18220 10](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18220-10.png)
Question 2.
Distance between foci = 2ae
Given, distance between foci = 8
2ae = 8
2(5)e = 8
(vi) Given, the length of the latus rectum is 6, and co-ordinates of foci are (±2, 0).
The foci of the ellipse are on the X-axis.
The ellipse passes through the points (-3, 1) and (2, -2).
Substituting x = -3 and y = 1 in equation of ellipse, we get
Equations (i) and (ii) become
9A + B = 1 …..(iii)
4A + 4B = 1 …..(iv)
Multiplying (iii) by 4, we get
36A + 4B = 4 …..(v)
Subtracting (iv) from (v), we get
32A = 3
The ellipse passes through (-√5, 2).
Substituting x = -√5 and y = 2 in equation of ellipse, we get
![Maharashtra Board Class 11 Maths Part 1 Chapter 7 Conic Sections Ex 7.2 Solution 30 word image 18220 30](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18220-30.png)
![Maharashtra Board Class 11 Maths Part 1 Chapter 7 Conic Sections Ex 7.2 Solution 31 word image 18220 31](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18220-31.jpeg)
![Maharashtra Board Class 11 Maths Part 1 Chapter 7 Conic Sections Ex 7.2 Solution 32 word image 18220 32](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18220-32.png)
Question 3.
Find the eccentricity of an ellipse, if the length of its latus rectum is one-third of its minor axis.
Solution:
Question 4.
Find the eccentricity of an ellipse, if the distance between its directrices is three times the distance between its foci.
Solution:
Question 5.
Question 6.
Question 7.
Question 8.
Question 9.
Question 10.
Question 11.
Solution:
![Maharashtra Board Class 11 Maths Part 1 Chapter 7 Conic Sections Ex 7.2 Solution 49 word image 18220 50](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18220-50.png)
![Maharashtra Board Class 11 Maths Part 1 Chapter 7 Conic Sections Ex 7.2 Solution 50 word image 18220 51](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18220-51.png)
![Maharashtra Board Class 11 Maths Part 1 Chapter 7 Conic Sections Ex 7.2 Solution 51 word image 18220 52](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18220-52.png)
![Maharashtra Board Class 11 Maths Part 1 Chapter 7 Conic Sections Ex 7.2 Solution 52 word image 18220 53](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18220-53.jpeg)
Slope of the given line x + y + 1 = 0 is -1.
Since the given line is parallel to the required tangents,
the slope of the required tangents is m = -1.
Equations of tangents to the ellipse
![Maharashtra Board Class 11 Maths Part 1 Chapter 7 Conic Sections Ex 7.2 Solution 55 word image 18220 56](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18220-56.png)
![Maharashtra Board Class 11 Maths Part 1 Chapter 7 Conic Sections Ex 7.2 Solution 56 word image 18220 57](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18220-57.jpeg)
![Maharashtra Board Class 11 Maths Part 1 Chapter 7 Conic Sections Ex 7.2 Solution 57 word image 18220 58](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/word-image-18220-58.png)
![Maharashtra Board Class 11 Maths Part 1 Chapter 7 Conic Sections Ex 7.2 Solution 58 Maharashtra Board 11th Maths Solutions Chapter 7 Conic Sections Ex 7.2 Q11 (vii)](https://mhboardsolutions.xyz/wp-content/uploads/2022/02/maharashtra-board-11th-maths-solutions-chapter-7-c-12.png)
Question 12.
Alternate method:
The locus of the point of intersection of perpendicular tangents is the director circle of an ellipse.
Question 13.
Question 14.
Show that the locus of the point of intersection of tangents at two points on an ellipse, whose eccentric angles differ by a constant, is an ellipse.
Solution:
Question 15.
Question 16.
Question 17.
Question 18.