Question 23

If $$\frac{a}{b+c}+\frac{c}{a+b}=\frac{2b}{c+a}$$, where $$a+b$$, $$b+c$$, $$c+a$$ and $$a+b+c$$ are all nonzero, then the numerical value of $$\frac{a^2+c^2}{b^2}$$ is

Correct Answer: 2

Solution

Multiplying the given equation by $$(a+b)(b+c)(c+a)$$ and simplifying gives $$(a+b+c)(a^2-2b^2+c^2)=0$$. Since $$a+b+c$$ is nonzero, we must have $$a^2+c^2=2b^2$$. Therefore $$\frac{a^2+c^2}{b^2}=2$$.

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