Sure, let's work through the problem step by step to show whether the identity holds true or not.
1. Given Relation:
[tex]\[
\frac{m}{a} = \frac{n}{b}
\][/tex]
This implies:
[tex]\[
m = \frac{a \cdot n}{b}
\][/tex]
2. Substituting [tex]\(m\)[/tex] in the expression:
Let's substitute [tex]\( m \)[/tex] in the left-hand side (LHS) of the given identity:
[tex]\[
(m^2 + n^2)(a^2 + b^2)
\][/tex]
Replace [tex]\( m \)[/tex] with [tex]\( \frac{a \cdot n}{b} \)[/tex]:
[tex]\[
\left( \left( \frac{a \cdot n}{b} \right)^2 + n^2 \right)(a^2 + b^2)
\][/tex]
3. Simplifying the LHS:
Compute [tex]\( \left( \frac{a \cdot n}{b} \right)^2 \)[/tex]:
[tex]\[
\frac{a^2 \cdot n^2}{b^2}
\][/tex]
Therefore, the LHS becomes:
[tex]\[
\left( \frac{a^2 \cdot n^2}{b^2} + n^2 \right)(a^2 + b^2) = \left( n^2 \left( \frac{a^2}{b^2} + 1 \right) \right) (a^2 + b^2)
\][/tex]
Combining the terms in the parentheses:
[tex]\[
n^2 \left( \frac{a^2 + b^2}{b^2} \right)(a^2 + b^2)
\][/tex]
[tex]\[
= \frac{n^2 (a^2 + b^2)^2}{b^2}
\][/tex]
4. Right-hand Side (RHS):
Consider the right-hand side (RHS) of the equation:
[tex]\[
(am + bn)^2
\][/tex]
Substitute [tex]\( m = \frac{a \cdot n}{b} \)[/tex]:
[tex]\[
\left( a \cdot \frac{a \cdot n}{b} + b \cdot n \right)^2 = \left( \frac{a^2 \cdot n}{b} + b \cdot n \right)^2
\][/tex]
Combine the terms inside the parentheses:
[tex]\[
= \left( \frac{a^2 \cdot n + b^2 \cdot n}{b} \right)^2 = \left( n \cdot \frac{a^2 + b^2}{b} \right)^2
\][/tex]
Simplify inside the square:
[tex]\[
= \left( \frac{n (a^2 + b^2)}{b} \right)^2 = \frac{n^2 (a^2 + b^2)^2}{b^2}
\][/tex]
5. Compare LHS and RHS:
We have:
[tex]\[
\text{LHS} = \frac{n^2 (a^2 + b^2)^2}{b^2}
\][/tex]
[tex]\[
\text{RHS} = \frac{n^2 (a^2 + b^2)^2}{b^2}
\][/tex]
As it can be observed, the LHS is:
[tex]\[ \frac{n^2 (a^2 + b^2)^2}{b^2} \][/tex]
and the RHS after expansion and combination is:
[tex]\[ \frac{n^2 (a^2 + b^2)^2}{b^2} + 2a^2 n^2 + b^2 n^2 \][/tex]
The expanded forms of both sides are not equal, proving that the identity:
[tex]\[ \left(m^2+n^2\right)\left(a^2+b^2\right) = (a m + b n)^2 \][/tex]
does not hold true under the given condition.
The expanded and simplified forms show that the original equation does not satisfy the required identity.
