If (4x2 + xy) : (3xy - y2) = 12 : 5, find (x + 2y) : (2x + y).
Answers
Answer:
(i) (4x 2 +xy):(3xy–y 2 )=12:5
Step-by-step explanation:
We can write it as
(3xy–y
2
)
(4x
2
+xy)
=
5
12
By cross multiplication
20x
2
+5xy=36xy–12y
2
20x
2
+5xy–36xy+12y
2
=0
20x
2
–31xy+12y
2
=0
Now divide the entire equation by y
2
20
y
2
x
2
–31
y
2
xy
+12
y
2
y
2
=0
So we get
20(
y
x
)
2
–31(
y
x
)+12=0
20(
y
x
)
2
–15(
y
x
)−16(
y
x
)+12=0
Taking common terms
5(
y
x
)[4(
y
x
)–3]–4[4(
y
x
)–3]=0
[4(
y
x
)–3]×[5(
y
x
)–4]=0
Here 4(
y
x
)–3=0 or 5(
y
x
)−4=0
4(
y
x
)=3
So we get
y
x
=
4
3
Similarly 5(
y
x
)=4
So we get
y
x
=
5
4
Now dividing given equation by y
(2x+y)
(x+2y)
=
(2
y
x
+1)
(
y
x
+2)
(a) If
y
x
=
4
3
, then
Substituting the values
(2x+y)
(x+2y)
=
(2×
4
3
+1)
(
4
3
+2)
By further calculation
=
2
5
4
11
=
4
11
×
5
2
=
10
11
So we get
(x+2y):(2x+y)=11:10
(b) If
y
x
=
5
4
then
(2x+y)
(x+2y)
=
[2
y
x
+1]
[
y
x
+2]
Substituting the value of
y
x
(2x+y)
(x+2y)
=
[2×
5
4
+1]
[
5
4
+2]
So we get
(2x+y)
(x+2y)
=
5
13
5
14
13
14
So, we get
(2x+y)
(x+2y)
=
10
11
or
13
14
(x+2y):(2x+y)=11:10 or 14:13
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