Question

if a2+b2=5ab then show a2/b2+b2/a2=23

Answer 1

a2/b2+b2/a2=(a4+b4)/a2b2
=[(a2+b2)^2-2a2b2]/a2b2
=[(5ab)^2-2a2b2]/a2b2
=[25a^2b^2-2a2b2]/a2b2
=23a2b2/a2b2
=23

Answer 2

Answer:

23

Step-by-step explanation:

Given:

$a^{2}+b^{2}=5 a b$

To prove:

$\frac{a^{2}}{b^{2}}+\frac{b^{2}}{a^{2}}=23$

Solution:

A quadratic equation can be solved using one of two basic strategies. Those are

The method of standard equations

Factorization is a technique.

The roots of a quadratic equation can be discovered using the Standard equation approach.

$\begin{aligned}&a^{2}+b^{2}=5 a b \\&\Rightarrow \frac{a^{2}}{a b}+\frac{b^{2}}{a b}=5 \\&\Rightarrow \frac{a}{b}+\frac{b}{a}=5\end{aligned}$

Now squaring the equation on both sides,

$\begin{aligned}&\Rightarrow\left(\frac{a}{b}\right)^{2}+\left(\frac{b}{a}\right)^{2}+2 \times \frac{a}{b} \times \frac{b}{a}=25 \\&\Rightarrow \frac{a^{2}}{b^{2}}+\frac{b^{2}}{a^{2}}=25-2 \\&\Rightarrow \frac{a^{2}}{b^{2}}+\frac{b^{2}}{a^{2}}=23\end{aligned}$

Hence the correct answer is 23.

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