NCERT Solutions for Class 11 Maths Chapter 11 – Conic Sections Ex 11.3
Page No 255:
Question 1:
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5443/chapter%2011_html_44e3930c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5443/chapter%2011_html_44e3930c.gif)
Answer:
The given equation is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5443/chapter%2011_html_44e3930c.gif)
Here, the denominator of
is greater than the denominator of
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5443/chapter%2011_html_86db3.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5443/chapter%2011_html_5d4ada9a.gif)
Therefore, the major axis is along the x-axis, while the minor axis is along the y-axis.
On comparing the given equation with
, we obtain a = 6 and b = 4.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5443/chapter%2011_html_m64d080ba.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5443/chapter%2011_html_2c5a2d4c.gif)
Therefore,
The coordinates of the foci are
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5443/chapter%2011_html_m578ee8b6.gif)
The coordinates of the vertices are (6, 0) and (–6, 0).
Length of major axis = 2a = 12
Length of minor axis = 2b = 8
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5443/chapter%2011_html_m2b5bc57.gif)
Length of latus rectum ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5443/chapter%2011_html_m76de02aa.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5443/chapter%2011_html_m76de02aa.gif)
Question 2:
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5444/chapter%2011_html_10a4843f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5444/chapter%2011_html_10a4843f.gif)
Answer:
The given equation is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5444/chapter%2011_html_m74596571.gif)
Here, the denominator of
is greater than the denominator of
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5444/chapter%2011_html_2c2847ad.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5444/chapter%2011_html_6d70d3c8.gif)
Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.
On comparing the given equation with
, we obtain b = 2 and a = 5.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5444/chapter%2011_html_m4d428e43.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5444/chapter%2011_html_m123b0d3a.gif)
Therefore,
The coordinates of the foci are
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5444/chapter%2011_html_6065d9f0.gif)
The coordinates of the vertices are (0, 5) and (0, –5)
Length of major axis = 2a = 10
Length of minor axis = 2b = 4
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5444/chapter%2011_html_29949993.gif)
Length of latus rectum ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5444/chapter%2011_html_45386b26.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5444/chapter%2011_html_45386b26.gif)
Question 3:
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5445/chapter%2011_html_b40375c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5445/chapter%2011_html_b40375c.gif)
Answer:
The given equation is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5445/chapter%2011_html_18135d87.gif)
Here, the denominator of
is greater than the denominator of
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5445/chapter%2011_html_9f5a351.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5445/chapter%2011_html_m3668c715.gif)
Therefore, the major axis is along the x-axis, while the minor axis is along the y-axis.
On comparing the given equation with
, we obtain a = 4 and b = 3.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5445/chapter%2011_html_m64d080ba.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5445/chapter%2011_html_2c16e717.gif)
Therefore,
The coordinates of the foci are
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5445/chapter%2011_html_m53b00781.gif)
The coordinates of the vertices are
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5445/chapter%2011_html_686926ab.gif)
Length of major axis = 2a = 8
Length of minor axis = 2b = 6
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5445/chapter%2011_html_m68a90d39.gif)
Length of latus rectum ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5445/chapter%2011_html_3818a1a6.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5445/chapter%2011_html_3818a1a6.gif)
Question 4:
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5446/chapter%2011_html_m1a8440c4.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5446/chapter%2011_html_m1a8440c4.gif)
Answer:
The given equation is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5446/chapter%2011_html_m1fde515.gif)
Here, the denominator of
is greater than the denominator of
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5446/chapter%2011_html_m3a0b37d5.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5446/chapter%2011_html_27cf422a.gif)
Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.
On comparing the given equation with
, we obtain b = 5 and a = 10.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5446/chapter%2011_html_m4d428e43.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5446/chapter%2011_html_m68f80cee.gif)
Therefore,
The coordinates of the foci are
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5446/chapter%2011_html_68df6116.gif)
The coordinates of the vertices are (0, ±10).
Length of major axis = 2a = 20
Length of minor axis = 2b = 10
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5446/chapter%2011_html_m39be10d1.gif)
Length of latus rectum ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5446/chapter%2011_html_m4f49d33f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5446/chapter%2011_html_m4f49d33f.gif)
Question 5:
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5447/chapter%2011_html_50332c76.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5447/chapter%2011_html_50332c76.gif)
Answer:
The given equation is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5447/chapter%2011_html_m3e91ef85.gif)
Here, the denominator of
is greater than the denominator of
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5447/chapter%2011_html_m265f42b7.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5447/chapter%2011_html_m2f4628be.gif)
Therefore, the major axis is along the x-axis, while the minor axis is along the y-axis.
On comparing the given equation with
, we obtain a = 7 and b = 6.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5447/chapter%2011_html_m64d080ba.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5447/chapter%2011_html_234dc835.gif)
Therefore,
The coordinates of the foci are
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5447/chapter%2011_html_m2b5a897e.gif)
The coordinates of the vertices are (± 7, 0).
Length of major axis = 2a = 14
Length of minor axis = 2b = 12
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5447/chapter%2011_html_2d5f65b0.gif)
Length of latus rectum![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5447/chapter%2011_html_62479d1e.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5447/chapter%2011_html_62479d1e.gif)
Question 6:
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5448/chapter%2011_html_5983e98d.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5448/chapter%2011_html_5983e98d.gif)
Answer:
The given equation is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5448/chapter%2011_html_m26bb6d42.gif)
Here, the denominator of
is greater than the denominator of
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5448/chapter%2011_html_446b35d9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5448/chapter%2011_html_fce83d.gif)
Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.
On comparing the given equation with
, we obtain b = 10 and a = 20.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5448/chapter%2011_html_m4d428e43.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5448/chapter%2011_html_7ec749ac.gif)
Therefore,
The coordinates of the foci are
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5448/chapter%2011_html_m3fb3a3e4.gif)
The coordinates of the vertices are (0, ±20)
Length of major axis = 2a = 40
Length of minor axis = 2b = 20
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5448/chapter%2011_html_7889be10.gif)
Length of latus rectum![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5448/chapter%2011_html_m2d25d1d9.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5448/chapter%2011_html_m2d25d1d9.gif)
Question 7:
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 36x2 + 4y2 = 144
Answer:
The given equation is 36x2 + 4y2 = 144.
It can be written as
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5449/chapter%2011_html_1e69b416.gif)
Here, the denominator of
is greater than the denominator of
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5449/chapter%2011_html_69710dc1.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5449/chapter%2011_html_m57101b8d.gif)
Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.
On comparing equation (1) with
, we obtain b = 2 and a = 6.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5449/chapter%2011_html_m4d428e43.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5449/chapter%2011_html_m153541ab.gif)
Therefore,
The coordinates of the foci are
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5449/chapter%2011_html_m31ce93e4.gif)
The coordinates of the vertices are (0, ±6).
Length of major axis = 2a = 12
Length of minor axis = 2b = 4
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5449/chapter%2011_html_m56527a1b.gif)
Length of latus rectum ![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5449/chapter%2011_html_5d41d0a0.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5449/chapter%2011_html_5d41d0a0.gif)
Question 8:
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 16x2 + y2 = 16
Answer:
The given equation is 16x2 + y2 = 16.
It can be written as
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5450/chapter%2011_html_mfdf949d.gif)
Here, the denominator of
is greater than the denominator of
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5450/chapter%2011_html_1355c9ae.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5450/chapter%2011_html_78195c51.gif)
Therefore, the major axis is along the y-axis, while the minor axis is along the x-axis.
On comparing equation (1) with
, we obtain b = 1 and a = 4.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5450/chapter%2011_html_m4d428e43.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5450/chapter%2011_html_m497b97f2.gif)
Therefore,
The coordinates of the foci are
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5450/chapter%2011_html_m44c45a02.gif)
The coordinates of the vertices are (0, ±4).
Length of major axis = 2a = 8
Length of minor axis = 2b = 2
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5450/chapter%2011_html_5ec1d73d.gif)
Length of latus rectum![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5450/chapter%2011_html_4a0fc915.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5450/chapter%2011_html_4a0fc915.gif)
Question 9:
Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse 4x2 + 9y2 = 36
Answer:
The given equation is 4x2 + 9y2 = 36.
It can be written as
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5451/chapter%2011_html_m6f85a24e.gif)
Here, the denominator of
is greater than the denominator of
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5451/chapter%2011_html_m5c172c6f.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5451/chapter%2011_html_12025d58.gif)
Therefore, the major axis is along the x-axis, while the minor axis is along the y-axis.
On comparing the given equation with
, we obtain a = 3 and b = 2.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5451/chapter%2011_html_m64d080ba.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5451/chapter%2011_html_73f2d960.gif)
Therefore,
The coordinates of the foci are
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5451/chapter%2011_html_mf509437.gif)
The coordinates of the vertices are (±3, 0).
Length of major axis = 2a = 6
Length of minor axis = 2b = 4
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5451/chapter%2011_html_m3938b939.gif)
Length of latus rectum![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5451/chapter%2011_html_m7f673582.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5451/chapter%2011_html_m7f673582.gif)
Question 10:
Find the equation for the ellipse that satisfies the given conditions: Vertices (±5, 0), foci (±4, 0)
Answer:
Vertices (±5, 0), foci (±4, 0)
Here, the vertices are on the x-axis.
Therefore, the equation of the ellipse will be of the form
, where a is the semi-major axis.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5452/chapter%2011_html_m64d080ba.gif)
Accordingly, a = 5 and c = 4.
It is known that
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5452/chapter%2011_html_m3286f72c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5452/chapter%2011_html_530eb685.gif)
Thus, the equation of the ellipse is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5452/chapter%2011_html_361f5bc2.gif)
Question 11:
Find the equation for the ellipse that satisfies the given conditions: Vertices (0, ±13), foci (0, ±5)
Answer:
Vertices (0, ±13), foci (0, ±5)
Here, the vertices are on the y-axis.
Therefore, the equation of the ellipse will be of the form
, where a is the semi-major axis.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5453/chapter%2011_html_m4d428e43.gif)
Accordingly, a = 13 and c = 5.
It is known that
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5453/chapter%2011_html_m3286f72c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5453/chapter%2011_html_1f1643e0.gif)
Thus, the equation of the ellipse is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5453/chapter%2011_html_m7a10013.gif)
Question 12:
Find the equation for the ellipse that satisfies the given conditions: Vertices (±6, 0), foci (±4, 0)
Answer:
Vertices (±6, 0), foci (±4, 0)
Here, the vertices are on the x-axis.
Therefore, the equation of the ellipse will be of the form
, where a is the semi-major axis.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5454/chapter%2011_html_m64d080ba.gif)
Accordingly, a = 6, c = 4.
It is known that
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5454/chapter%2011_html_m3286f72c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5454/chapter%2011_html_m76a9ffa1.gif)
Thus, the equation of the ellipse is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5454/chapter%2011_html_m6d188d9d.gif)
Question 13:
Find the equation for the ellipse that satisfies the given conditions: Ends of major axis (±3, 0), ends of minor axis (0, ±2)
Answer:
Ends of major axis (±3, 0), ends of minor axis (0, ±2)
Here, the major axis is along the x-axis.
Therefore, the equation of the ellipse will be of the form
, where a is the semi-major axis.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5455/chapter%2011_html_m64d080ba.gif)
Accordingly, a = 3 and b = 2.
Thus, the equation of the ellipse is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5455/chapter%2011_html_m73a0b10d.gif)
Question 14:
Find the equation for the ellipse that satisfies the given conditions: Ends of major axis
, ends of minor axis (±1, 0)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5456/chapter%2011_html_m716c9d36.gif)
Answer:
Ends of major axis
, ends of minor axis (±1, 0)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5456/chapter%2011_html_m716c9d36.gif)
Here, the major axis is along the y-axis.
Therefore, the equation of the ellipse will be of the form
, where a is the semi-major axis.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5456/chapter%2011_html_m4d428e43.gif)
Accordingly, a =
and b = 1.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5456/chapter%2011_html_52caafc6.gif)
Thus, the equation of the ellipse is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5456/chapter%2011_html_m73e00e5a.gif)
Question 15:
Find the equation for the ellipse that satisfies the given conditions: Length of major axis 26, foci (±5, 0)
Answer:
Length of major axis = 26; foci = (±5, 0).
Since the foci are on the x-axis, the major axis is along the x-axis.
Therefore, the equation of the ellipse will be of the form
, where a is the semi-major axis.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5457/chapter%2011_html_m64d080ba.gif)
Accordingly, 2a = 26 ⇒ a = 13 and c = 5.
It is known that
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5457/chapter%2011_html_m3286f72c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5457/chapter%2011_html_1f1643e0.gif)
Thus, the equation of the ellipse is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5457/chapter%2011_html_2e3e969d.gif)
Question 16:
Find the equation for the ellipse that satisfies the given conditions: Length of minor axis 16, foci (0, ±6)
Answer:
Length of minor axis = 16; foci = (0, ±6).
Since the foci are on the y-axis, the major axis is along the y-axis.
Therefore, the equation of the ellipse will be of the form
, where a is the semi-major axis.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5458/chapter%2011_html_m4d428e43.gif)
Accordingly, 2b = 16 ⇒ b = 8 and c = 6.
It is known that
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5458/chapter%2011_html_m3286f72c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5458/chapter%2011_html_m5bfd9fd4.gif)
Thus, the equation of the ellipse is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5458/chapter%2011_html_246d9d7b.gif)
Question 17:
Find the equation for the ellipse that satisfies the given conditions: Foci (±3, 0), a = 4
Answer:
Foci (±3, 0), a = 4
Since the foci are on the x-axis, the major axis is along the x-axis.
Therefore, the equation of the ellipse will be of the form
, where a is the semi-major axis.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5459/chapter%2011_html_m64d080ba.gif)
Accordingly, c = 3 and a = 4.
It is known that
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5459/chapter%2011_html_m3286f72c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5459/chapter%2011_html_68aa1337.gif)
Thus, the equation of the ellipse is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5459/chapter%2011_html_5effa4cf.gif)
Question 18:
Find the equation for the ellipse that satisfies the given conditions: b = 3, c = 4, centre at the origin; foci on the x axis.
Answer:
It is given that b = 3, c = 4, centre at the origin; foci on the x axis.
Since the foci are on the x-axis, the major axis is along the x-axis.
Therefore, the equation of the ellipse will be of the form
, where a is the semi-major axis.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5460/chapter%2011_html_m64d080ba.gif)
Accordingly, b = 3, c = 4.
It is known that
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5460/chapter%2011_html_m3286f72c.gif)
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5460/chapter%2011_html_66038583.gif)
Thus, the equation of the ellipse is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5460/chapter%2011_html_361f5bc2.gif)
Question 19:
Find the equation for the ellipse that satisfies the given conditions: Centre at (0, 0), major axis on the y-axis and passes through the points (3, 2) and (1, 6).
Answer:
Since the centre is at (0, 0) and the major axis is on the y-axis, the equation of the ellipse will be of the form
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5461/chapter%2011_html_m5d4fb08b.gif)
The ellipse passes through points (3, 2) and (1, 6). Hence,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5461/chapter%2011_html_aea257e.gif)
On solving equations (2) and (3), we obtain b2 = 10 and a2 = 40.
Thus, the equation of the ellipse is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5461/chapter%2011_html_7808e979.gif)
Question 20:
Find the equation for the ellipse that satisfies the given conditions: Major axis on the x-axis and passes through the points (4, 3) and (6, 2).
Answer:
Since the major axis is on the x-axis, the equation of the ellipse will be of the form
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5462/chapter%2011_html_m3c832535.gif)
The ellipse passes through points (4, 3) and (6, 2). Hence,
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5462/chapter%2011_html_m59b2810a.gif)
On solving equations (2) and (3), we obtain a2 = 52 and b2 = 13.
Thus, the equation of the ellipse is
.
![](https://img-nm.mnimgs.com/img/study_content/curr/1/11/11/171/5462/chapter%2011_html_4eddb78.gif)