if 15a²+4b²÷15a²-4b² =47÷7 then find the value of b³-2a³÷b³+2a³
Answers
Answer:
7a+3b
7a−3b
=
6
1
; if a>0
7a+3b
7a−3b
=6 ; if a<0
Step-by-step explanation:
Given:
\rm \dfrac{15a^2+4b^2}{15a^2-4b^2}=\dfrac {47}{7}
15a
2
−4b
2
15a
2
+4b
2
=
7
47
The Componendo and Dividendo theorem states that if there is a fraction such as,
\rm \dfrac {N^m}{D^m}=\dfrac pq
D
m
N
m
=
q
p
,
then,
\rm \dfrac{N^m+D^m}{N^m-D^m}=\dfrac{p+q}{p-q}
N
m
−D
m
N
m
+D
m
=
p−q
p+q
.
Applying this rule to the given equation by putting,
\begin{gathered}\rm N^m= 15a^2+4b^2\\D^m=15a^2-4b^2\\p=47\\q=7\end{gathered}
N
m
=15a
2
+4b
2
D
m
=15a
2
−4b
2
p=47
q=7
we get,
\begin{gathered}\rm \dfrac{(15a^2+4b^2)+(15a^2-4b^2)}{(15a^2+4b^2)-(15a^2-4b^2)}=\dfrac{47+7}{47-7}\\\dfrac{15a^2+15a^2+4b^2-4b^2}{15a^2-15a^2+4b^2-(-4b^2)}=\dfrac{54}{40}\\\dfrac{30a^2}{8b^2} =\dfrac{27}{20}\\\dfrac{a^2}{b^2} = \dfrac {8}{30} \times \dfrac{27}{20} = \dfrac {9}{25}\\\Rightarrow \dfrac ab=\pm \sqrt{\dfrac{9}{25}} = \pm \dfrac 35\end{gathered}
(15a
2
+4b
2
)−(15a
2
−4b
2
)
(15a
2
+4b
2
)+(15a
2
−4b
2
)
=
47−7
47+7
15a
2
−15a
2
+4b
2
−(−4b
2
)
15a
2
+15a
2
+4b
2
−4b
2
=
40
54
8b
2
30a
2
=
20
27
b
2
a
2
=
30
8
×
20
27
=
25
9
⇒
b
a
=±
25
9
=±
5
3
Therefore,
\rm a = \pm \dfrac 35 ba=±
5
3
b
Taking positive value of a.
\begin{gathered}\rm a=\dfrac 35 b\\\\\dfrac{7a-3b}{7a+3b}=\dfrac{7\left( \dfrac 35 b\right) -3b}{7\left( \dfrac 35 b\right) +3b}\\=\dfrac{\dfrac {21b}{5}-3b}{\dfrac {21b}{5} +3b}\\=\dfrac{\dfrac{21b-15b}{5}}{\dfrac{21b+15b}{5}}\\=\dfrac{6b}{36b}\\=\dfrac 16\end{gathered}
a=
5
3
b
7a+3b
7a−3b
=
7(
5
3
b)+3b
7(
5
3
b)−3b
=
5
21b
+3b
5
21b
−3b
=
5
21b+15b
5
21b−15b
=
36b
6b
=
6
1
Taking negative value of a.
\begin{gathered}\rm a=-\dfrac 35 b\\\\\dfrac{7a-3b}{7a+3b}=\dfrac{7\left( -\dfrac 35 b\right) -3b}{7\left( -\dfrac 35 b\right) +3b}\\=\dfrac{-\dfrac {21b}{5}-3b}{-\dfrac {21b}{5} +3b}\\=\dfrac{\dfrac{-21b-15b}{5}}{\dfrac{-21b+15b}{5}}\\=\dfrac{-36b}{-6b}\\=6.\end{gathered}
a=−
5
3
b
7a+3b
7a−3b
=
7(−
5
3
b)+3b
7(−
5
3
b)−3b
=
−
5
21b
+3b
−
5
21b
−3b
=
5
−21b+15b
5
−21b−15b
=
−6b
−36b
=6.