** ****Q.1: Obtain the coordinates of the vertices, the foci, the major and the minor axis, the length of the latus rectum and the eccentricity of the ellipse x225+y29=1.**

** **

**Sol:**

**Given:**

**. . . . . . . . (1) is the equation of the ellipse.**

As we know that the **denominator** of

So, along x – axis consist the major axis and along the y – axis consists the minor axis.

**. . . . . . . . . . . . . . (2)**

**Now, on comparing equation (1) and equation (2) we will get:**

We get:

**m = 5, n = 3**

The **vertices** **coordinates** are (m, 0) and (- m, 0) **= (5, 0), (- 5, 0)**

The **foci’s** **coordinates** are (c, 0) and (- c, 0) **= (4, 0) and (- 4, 0)**

Length of the axis:

**Major axis** = 2m **= 10**

**Minor axis** = 2n **= 6**

**Length of the latus rectum =**

**Eccentricity, e =**

**Q.2: Obtain the coordinates of the vertices, the foci, the major and the minor axis, the length of the latus rectum and the eccentricity of the ellipse x29+y216=1.**

** **

**Sol:**

**Given:**

**. . . . . . . . . . (1) is the equation of the ellipse.**

As we know that the **denominator** of

So, along y – axis consist the major axis and along the x – axis consists the minor axis.

**. . . . . . . . . . . . (2)**

**Now, on comparing equation (1) and equation (2) we will get:**

We get,:

**m = 4, n = 3**

The **vertices coordinates** are (0, m) and (0, – m) **= (0, 4), (0, – 4)**

The **foci’s coordinates** are (0, c) and (0, – c)** = ( 7–√, 0) and (-7–√, 0)**

**Length of the axis:**

**Major axis** = 2m **= 8**

**Minor axis** = 2n **= 6**

**Length of the latus rectum =**

**Eccentricity, e =**

**Q.3: Obtain the coordinates of the vertices, the foci, the major and the minor axis, the length of the latus rectum and the eccentricity of the ellipse x225+y24=1.**

** **

**Sol:**

**Given:**

**(1) is the equation of the ellipse.**

As we know that the **denominator** of

So, along x – axis consist the major axis and along the y – axis consists the minor axis.

**. . . . . . . . . . (2)**

**Now, on comparing equation (1) and equation (2) we will get:**

We get:

**m = 5, n = 2**

The **vertices** **coordinates** are (m, 0) and (- m, 0) **= (5, 0), (- 5, 0)**

The **foci’s coordinates** are (c, 0) and (- c, 0) **= ( 21−−√, 0) and (-21−−√, 0)**

**Length of the axis:**

**Major axis =** 2m = **10**

**Minor axis =** 2n = **4**

**Length of the latus rectum =**

**Eccentricity, e =**

**Q.4: Obtain the coordinates of the vertices, the foci, the major and the minor axis, the length of the latus rectum and the eccentricity of the ellipse x236+y2121=1.**

** **

**Sol:**

**Given:**

**. . . . . . . . . . (1) is the equation of the ellipse.**

As we know that the **denominator** of

So, along y – axis consist the major axis and along the x – axis consists the minor axis.

**. . . . . . . . . . . . . . . (2)**

**Now, on comparing equation (1) and equation (2) we will get:**

We get:

**m = 11, n = 6**

The **vertices coordinates** are (0, m) and (0, – m) **= (0, 11), (0, – 11)**

The **foci’s coordinates** are (0, c) and (0, – c) **= (0, 85−−√) and (0, –85−−√)**

**Length of the axis:**

**Major axis =** 2m **= 22**

**Minor axis =** 2n **= 12**

Length of the **latus rectum** =

**Eccentricity, e =** **= 85√11**

**Q.5: Obtain the coordinates of the vertices, the foci, the major and the minor axis, the length of the latus rectum and the eccentricity of the ellipse x264+y225=1.**

** **

**Sol:**

**Given:**

**. . . . . . . . . . . . . (1) is the equation of the ellipse.**

As we know that the **denominator** of

So, along x – axis consist the major axis and along the y – axis consists the minor axis.

**. . . . . . . . . . . . . (2)**

**Now, on comparing equation (1) and equation (2) we will get:**

We get:

**m = 8, n = 5**

The **vertices coordinates** are (m, 0) and (- m, 0) **= (8, 0), (- 8, 0)**

The **foci’s coordinates **are (c, 0) and (- c, 0) **= ( 39−−√, 0) and (-39−−√, 0)**

**Length of the axis:**

**Major axis** = 2m **= 16**

**Minor axis** = 2n **= 10**

Length of the **latus rectum** =

**Eccentricity, e** = **= 39√8**

**Q.6: Obtain the coordinates of the vertices, the foci, the major and the minor axis, the length of the latus rectum and the eccentricity of the ellipse x2400+y2100=1.**

** **

**Sol:**

**Given:**

** . . . . . . . . . . . . (1) is the equation of the ellipse.**

As we know that the **denominator** of

So, along x – axis consist the major axis and along the y – axis consists the minor axis.

**. . . . . . . . . . . (2)**

**Now, on comparing equation (1) and equation (2) we will get:**

We get:

**m = 20, n = 10**

The **vertices coordinates** are (m, 0) and (- m, 0) **= (20, 0), (- 20, 0)**

The **foci’s coordinates** are (c, 0) and (- c, 0) **= ( 103–√, 0) and (-103–√, 0)**

**Length of the axis:**

**Major axis** = 2m **= 40**

**Minor axis** = 2n **= 20**

**Length of the latus rectum** =

**Eccentricity, e** =

**Q.7: Obtain the coordinates of the vertices, the foci, the major and the minor axis, the length of the latus rectum and the eccentricity of the ellipse 49x ^{2} + 9y^{2 }= 441**

** **

**Sol:**

**Given:**

**49x ^{2} + 9y^{2 }= 441**

** . . . . . . . . . . . . . . . (1) is the equation of the ellipse.**

As we know that the **denominator** of

So, along y – axis consist the major axis and along the x – axis consists the minor axis.

**. . . . . . . . . . . . . . . (2)**

**Now, on comparing equation (1) and equation (2) we will get:**

We get:

**m = 7, n = 3**

The **vertices coordinates** are (0, m) and (0, – m) **= (0, 7), (0, – 7)**

The **foci’s coordinates** are (0, c) and (0, – c) **= (0, 210−−√) and (0, 210−−√)**

**Length of the axis:**

**Major axis **= 2m** = 14**

**Minor axis** = 2n **= 6**

**Length of the** **latus rectum =**

**Eccentricity, e =**

**Q.8: Obtain the coordinates of the vertices, the foci, the major and the minor axis, the length of the latus rectum and the eccentricity of the ellipse 4x ^{2} + 16y^{2 }= 64**

** **

**Sol:**

**Given:**

**4x ^{2} + 16y^{2 }= 64**

** . . . . . . . . . . . . . . . . . (1) is the equation of the ellipse.**

As we know that the **denominator** of

So, along x – axis consist the major axis and along the y – axis consists the minor axis.

** . . . . . . . . . . . . . . . (2)**

**Now, on comparing equation (1) and equation (2) we will get:**

We get:

**m = 4, n = 2**

The **vertices coordinates** are (m, 0) and (- m, 0) **= (4, 0), (- 4, 0)**

The **foci’s coordinates** are (c, 0) and (- c, 0) **= ( 23–√, 0) and (-23–√, 0)**

**Length of the axis:**

**Major axis** = 2m **= 8**

**Minor axis** = 2n **= 4**

**Length of the latus rectum =**

**Eccentricity, e =**

**Q.9: Obtain the coordinates of the vertices, the foci, the major and the minor axis, the length of the latus rectum and the eccentricity of the ellipse 4x ^{2} + y^{2 }= 4**

** **

**Sol:**

Given:

4x^{2} + y^{2 }= 4

**. . . . . . . . . . . . . . . . (1) is the equation of the ellipse.**

As we know that the **denominator** of

So, along y – axis consist the major axis and along the x – axis consists the minor axis.

**. . . . . . . . . . . . . (2)**

**Now, on comparing equation (1) and equation (2) we will get:**

We get:

**m = 2, n = 1**

The **vertices coordinates** are (0, m) and (0, – m) **= (0, 2), (0, – 2)**

The **foci’s coordinates** are (0, c) and (0, – c) **= (0, 3–√) and (0, –3–√)**

Length of the axis:

**Major axis** = 2m **= 4**

**Minor axis** = 2n **= 2**

**Length of the latus rectum =**

**Eccentricity, e =**

**Q.10: For the given condition obtain the equation of the ellipse.**

**(i) Vertices (****±****6, 0)**

**(ii) Foci (****±****3, 0)**

** **

**Sol:**

**Given:**

**(i) Vertices (****±****6, 0)**

**(ii) Foci (****±****3, 0)**

The vertices are represented along x – axis.

The required **equation of the ellipse **is of the form **. . . . . . . . . . . . . . . . . (1)**

We get:

**m = 6 (semi major axis), c = 3**

As we know:

**m ^{2} = n^{2} + c^{2}**

6^{2} = n^{2} + 3^{2}

n^{2} = 6^{2} – 3^{2}

n^{2} = 36 – 9 = 27

**n =**

**Hence, x236+y227=1 is the equation of the ellipse.**

**Q.11: For the given condition obtain the equation of the ellipse.**

**(i) Vertices (0, ****±****8)**

**(ii) Foci ( 0, ****±****4)**

** **

**Sol:**

**Given:**

**(i) Vertices (0, ****±****8)**

**(ii) Foci ( 0, ****±****4)**

The vertices are represented along y – axis.

The required **equation of the ellipse** is of the form **. . . . . . . . . . . . . . .(1)**

We get:

**m = 8 (semi major axis), c = 4**

As we know:

**m ^{2} = n^{2} + c^{2}**

8^{2} = n^{2} + 4^{2}

n^{2} = 8^{2} – 4^{2}

n^{2} = 64 – 16 = 48

**n =**

**Hence, x248+y264=1 is the equation of the ellipse.**

** **

**Q.12: For the given condition obtain the equation of the ellipse.**

**(i) Vertices (****±****5, 0)**

**(ii) Foci (****±****3, 0)**

** **

**Sol:**

**Given:**

**(i) Vertices (****±**** 5, 0)**

**(ii) Foci (****±**** 3, 0)**

The vertices are represented along x – axis.

The required **equation of the ellipse **is of the form **. . . . . . . . . . . . . . . (1)**

We get:

**m = 5 (semi major axis), c = 3**

As we know:

**m ^{2} = n^{2} + c^{2}**

5^{2} = n^{2} + 3^{2}

n^{2} = 5^{2} – 3^{2}

n^{2} = 25 – 9 = 16

**n =**

**Hence, x225+y216=1 is the equation of the ellipse.**

** **

**Q.13: For the given condition obtain the equation of the ellipse.**

**(i) Coordinates of major axis (****±****5, 0)**

**(ii) Coordinates of minor axis (0, ****±****3)**

** **

**Sol:**

**Given:**

**(i) Coordinates of major axis (****±****5, 0)**

**(ii) Coordinates of minor axis (0, ****±****3)**

The major axis is represented along x – axis.

The required **equation of the ellipse** is of the form

**. . . . . . . . . . . . . . . (1)**

**We get:**

**m = 5 (semi major axis), n = 3**

**Hence, x225+y216=1 is the equation of the ellipse.**

** **

**Q.14: For the given condition obtain the equation of the ellipse.**

**(i) Coordinates of major axis (0, ****±****2)**

**(ii) Coordinates of minor axis (****±**

** **

**Sol:**

**Given:**

**(i) Coordinates of major axis (0, ****±****2)**

**(ii) Coordinates of minor axis (****±**

The major axis is represented along y – axis.

The required **equation of the ellipse** is of the form **. . . . . . . . . . . . . . . . . (1)**

We get:

**m = 2 (semi major axis), n = 3–√**

**Hence, x23+y24=1 is the equation of the ellipse.**

** **

**Q.15: For the given condition obtain the equation of the ellipse.**

**(i) Length of major axis 30**

**(ii) Coordinates of foci (****±****4, 0)**

** **

**Sol:**

**Given:**

**(i) Length of major axis 30**

**(ii) Coordinates of foci (****±****4, 0)**

The major axis are represented along x – axis as foci is along x – axis

The required **equation of the ellipse** is of the form **. . . . . . . . . . . . . . . . . . . (1)**

We get:

2m = 30 (semi major axis),

**m = 15**

**c = 4**

As we know:

** m ^{2} = n^{2} + c^{2}**

15^{2} = n^{2} + 4^{2}

n^{2} = 15^{2} – 4^{2}

n^{2} = 225 – 16 = 209

**n =**

**Hence, x2225+y2209=1 is the equation of the ellipse.**

**Q.16: For the given condition obtain the equation of the ellipse.**

**(i) Length of minor axis 26**

**(ii) Coordinates of foci (0, ****±****9)**

** **

**Sol:**

**Given:**

**(i) Length of minor axis 26**

**(ii) Coordinates of foci (0, ****±****9)**

The major axis are represented along y – axis as foci is along y – axis

The required **equation of the ellipse** is of the form

**. . . . . . . . . . . . . (1)**

We get:

2n = 26 (semi major axis),

**n = 13**

**c = 9**

As we know:

**m ^{2} = n^{2} + c^{2}**

m^{2} = 13^{2} + 9^{2}

m^{2} = 169 + 81

m^{2} = 250

**m =**

**Hence, x2169+y2250=1 is the equation of the ellipse.**

** **

**Q.17: For the given condition obtain the equation of the ellipse.**

**(i) Coordinates of foci (****±****4, 0)**

**(ii) m = 6**

** **

**Sol:**

**Given:**

**(i) Coordinates of foci (****±****4, 0)**

**(ii) m = 6**

The major axis are represented along x – axis as foci is along x – axis

The required **equation of the ellipse** is of the form

**. . . . . . . . . . . . . . . (1)**

We get:

**c = 4**

**m = 6**

As we know:

**m ^{2} = n^{2} + c^{2}**

6^{2} = n^{2} + 4^{2}

n^{2} = 36 – 16

n^{2} = 20

**n =**

**Hence, x236+y220=1 is the equation of the ellipse.**

**Q.18: For the given condition obtain the equation of the ellipse.**

**(i) n = 2, c = 3 (on the x – axis)**

**(ii) Coordinates of centre (0, 0)**

** **

**Sol:**

**Given:**

**(i) n = 2, c = 3 (on the x – axis)**

**(ii) Coordinates of centre (0, 0) i.e., centre is at origin.**

The major axis are represented along x – axis as foci is along x – axis

The required **equation of the ellipse** is of the form:

**. . . . . . . . . . . . . (1)**

We have:

**n = 2, c = 3 **

As we know:

**m ^{2} = n^{2} + c^{2}**

m^{2} = 2^{2} + 3^{2}

m^{2} = 4 + 9

m^{2} = 13

**m =**

**Hence, x213+y24=1 is the equation of the ellipse.**

** **

**Q.19: For the given condition obtain the equation of the ellipse.**

**(i) Centre is at origin and major axis is along the y – axis**

**(ii) It passes through the points (1, 6) and (3, 2)**

** **

**Sol:**

**Given:**

**(i) Centre is at origin and major axis is along the y – axis**

**(ii) It passes through the points (1, 6) and (3, 2)**

The major axis are represented along y – axis as foci is along y – axis

The required **equation of the ellipse** is of the form:

**. . . . . . . . . . . . . . . (1)**

As the ellipse passes through (2, 1) and (2, 3) points, then

**. . . . . . . . . . . . . . . (a)**

**. . . . . . . . . . . . . . . . (b)**

**Now, on comparing equation (a) and equation (b) we will get:**

**m ^{2} = 40, n^{2} = 10**

**Hence, y240+x210=1 is the equation of the ellipse.**

**Q.20: For the given condition obtain the equation of the ellipse.**

**(i) Major axis is represented along x – axis**

**(ii) It passes through the points (6, 2) and (4, 3)**

** **

**Sol:**

**Given:**

**(i) Major axis is represented along x – axis**

**(ii) It passes through the points (6, 2) and (4, 3)**

The major axis are represented along y – axis as foci is along y – axis

The required **equation of the ellipse** is of the form:

**. . . . . . . . . . . . . . . . (1)**

As the ellipse passes through (6, 2) and (4, 3) points, then

**. . . . . . . . . . . . . . . . (a)**

**. . . . . . . . . . . . . . . . (b)**

**Now, on comparing equation (a) and equation (b) we will get:**

**m ^{2} = 52 and n^{2} = 13**

**Hence, y213+x252=1 is the equation of the ellipse.**