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Byju's Answer
Standard XII
Mathematics
Fundamental Laws of Logarithms
Find the cent...
Question
Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse:
(i) x
2
+ 2y
2
− 2x + 12y + 10 = 0
(ii) x
2
+ 4y
2
− 4x + 24y + 31 = 0
(iii) 4x
2
+ y
2
− 8x + 2y + 1 = 0
(iv) 3x
2
+ 4y
2
− 12x − 8y + 4 = 0
(v) 4x
2
+ 16y
2
− 24x − 32y − 12 = 0
(vi) x
2
+ 4y
2
− 2x = 0
Open in App
Solution
(
i
)
x
2
+
2
y
2
-
2
x
+
12
y
+
10
=
0
⇒
x
2
-
2
x
+
2
y
2
+
6
y
=
-
10
⇒
x
2
-
2
x
+
1
+
2
y
2
+
6
y
+
9
=
-
10
+
18
+
1
⇒
x
-
1
2
+
2
y
+
3
2
=
9
⇒
x
-
1
2
9
+
y
+
3
2
9
2
=
9
Here
,
x
1
=
1
a
n
d
y
1
=
-
3
Also
,
a
=
3
a
n
d
b
=
3
2
Centre=
1
,
-
3
Major axis=2
a
⇒
2
×
3
=
6
Minor axis=2
b
⇒
2
×
3
2
=
3
2
e
=
1
-
b
2
a
2
⇒
e
=
1
-
9
2
9
⇒
e
=
1
2
Foci
=
x
1
±
a
e
,
y
1
=
1
±
3
2
,
-
3
(
ii
)
x
2
+
4
y
2
-
4
x
+
24
y
+
31
=
0
⇒
x
2
-
4
x
+
4
y
2
+
6
y
=
-
31
⇒
x
2
-
4
x
+
4
+
4
y
2
+
6
y
+
9
=
-
31
+
36
+
4
⇒
x
-
2
2
+
4
y
+
3
2
=
9
⇒
x
-
2
2
9
+
y
+
3
2
9
4
=
9
Here
,
x
1
=
2
and
y
1
=
-
3
A
l
s
o
,
a
=
3
a
n
d
b
=
3
2
Centre=
x
1
,
y
1
=
2
,
-
3
Major axis=2
a
=
2
×
3
=
6
Minor axis=2
b
=
2
×
3
2
=
3
e
=
1
-
b
2
a
2
⇒
e
=
1
-
9
4
9
⇒
e
=
3
2
Foci
=
x
1
±
a
e
,
y
1
=
2
±
3
3
2
,
-
3
(
iii
)
4
x
2
+
y
2
-
8
x
+
2
y
+
1
=
0
⇒
4
x
2
-
2
x
+
y
2
+
2
y
=
-
1
⇒
4
x
2
-
2
x
+
1
+
y
2
+
2
y
+
1
=
-
1
+
4
+
1
⇒
4
x
-
1
2
+
y
+
1
2
=
4
⇒
x
-
1
2
1
+
y
+
1
2
4
=
1
Here
,
x
1
=
1
a
n
d
y
1
=
-
1
A
l
s
o
,
a
=
1
a
n
d
b
=
2
Centre=
x
1
,
y
1
=
1
,
-
1
Major axis=2
b
=
2
×
2
=
4
Minor axis=2
a
=
2
×
1
=
2
e
=
1
-
a
2
b
2
⇒
e
=
1
-
1
4
⇒
e
=
3
2
Foci
=
x
1
,
y
1
±
b
e
=
1
,
-
1
±
3
(
iv
)
3
x
2
+
4
y
2
-
12
x
-
8
y
+
4
=
0
⇒
3
x
2
-
4
x
+
4
y
2
-
2
y
=
-
4
⇒
3
x
2
-
4
x
+
4
+
4
y
2
-
2
y
+
1
=
-
4
+
12
+
4
⇒
3
x
-
2
2
+
4
y
-
1
2
=
12
⇒
x
-
2
2
4
+
y
-
1
2
3
=
1
Here
,
x
1
=
2
,
y
1
=
1
A
l
s
o
,
a
=
2
a
n
d
b
=
3
Centre=
x
1
,
y
1
=
2
,
1
Major axis=2
a
=
2
×
2
=
4
Minor axis=2
b
=
2
×
3
=
2
3
e
=
1
-
b
2
a
2
⇒
e
=
1
-
3
4
⇒
e
=
1
2
Foci
=
x
1
±
a
e
,
y
1
=
2
±
1
,
1
(
v
)
4
x
2
+
16
y
2
-
24
x
-
32
y
-
12
=
0
⇒
4
x
2
-
6
x
+
16
y
2
-
2
y
=
12
⇒
4
x
2
-
6
x
+
9
+
16
y
2
-
2
y
+
1
=
12
+
36
+
16
⇒
4
x
-
3
2
+
16
y
-
1
2
=
64
⇒
x
-
3
2
16
+
y
-
1
2
4
=
9
Centre=
3
,
1
M
ajor axis=2
a
=
2
×
4
=
8
Minor axis=2
b
=
2
×
2
=
4
e
=
1
-
b
2
a
2
⇒
e
=
1
-
4
16
⇒
e
=
3
2
Foci
=
x
±
a
e
,
y
=
3
±
2
3
,
1
(
vi
)
x
2
-
2
x
+
4
y
2
=
0
⇒
x
2
-
2
x
+
4
y
2
=
0
⇒
x
2
-
2
x
+
1
+
4
y
2
=
1
⇒
x
-
1
2
+
4
y
2
=
1
⇒
x
-
1
2
1
+
y
2
1
4
=
9
Centre=
1
,
0
Major axis=2
a
=
2
×
1
=
2
Minor axis=2
b
=
2
×
1
2
=
1
e
=
1
-
b
2
a
2
⇒
e
=
1
-
1
4
1
⇒
e
=
3
2
Foci
=
x
±
a
e
,
y
=
1
±
3
2
,
0
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0
Similar questions
Q.
Find the centre, the lengths of the axes, eccentricity, foci of the following ellipse:
(i)
x
2
+
2
y
2
−
2
x
+
12
y
+
10
=
0
(ii)
x
2
+
4
y
2
−
4
x
+
24
y
+
31
=
0
(iii)
4
x
2
+
y
2
−
8
x
+
2
y
+
1
=
0
(iv)
3
x
2
+
4
y
2
−
12
x
−
8
y
+
4
=
0
(v)
4
x
2
+
16
y
2
−
24
x
−
32
y
−
12
=
0
(vi)
x
2
+
4
y
2
−
2
x
=
0
Q.
Find the centre and radius of each of the circle
(i)
x
2
+
y
2
−
2
x
+
4
y
−
4
=
0
(ii)
4
x
2
+
4
y
2
−
24
x
−
8
y
−
24
=
0
Q.
Find the eccentricity, foci, centre, length of latus rectum vertices and the equation to the directrices of the hyperbola.
(a)
9
x
2
−
16
y
2
+
72
x
−
32
y
−
16
=
0
(b)
4
x
2
−
5
y
2
−
16
x
+
10
y
+
31
=
0
(c)
5
x
2
−
4
y
2
+
20
x
+
8
y
−
4
=
0
(d)
4
(
y
+
3
)
2
−
9
(
x
−
2
)
2
=
1
(e)
4
x
2
−
9
y
2
−
8
x
−
32
=
0
.
Q.
Find the values of a, b, c for following quadratic equations by comparing with standard form.
(i) x
2
+ 2x + 1 = 0
(ii) 2x
2
– x + 3 = 0
(iii) x
2
– x – 3 = 0
(iv) x
2
+ 5x – 4 = 0
(v) x
2
– 7x + 4 = 0
(vi) 4x
2
– 9x + 1 = 0
Q.
Find the eccentricity, coordinates of foci, length of the latus-rectum of the following ellipse:
(i) 4x
2
+ 9y
2
= 1
(ii) 5x
2
+ 4y
2
= 1
(iii) 4x
2
+ 3y
2
= 1
(iv) 25x
2
+ 16y
2
= 1600.
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