Question : If $\frac{a^{2} + b^{2}}{c^{2}} = \frac{b^{2} + c^{2}}{a^{2}} = \frac{c^{2} + a^{2}}{b^{2}} = \frac{1}{k}$ , $(k \neq 0)$ , then $k$ =?
Option 1: $2$
Option 2: $1$
Option 3: $0$
Option 4: $\frac{1}{2}$

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Question : If $a + \frac{1}{b} = b + \frac{1}{c} = c + \frac{1}{a}$ where $a \neq b \neq c \neq 0$ , then the value of $a^{2} b^{2} c^{2}$ is:

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Answer 1

Correct Answer: $\frac{1}{2}$

Solution : Given: $\frac{a^{2} + b^{2}}{c^{2}} = \frac{b^{2} + c^{2}}{a^{2}} = \frac{c^{2} + a^{2}}{b^{2}} = \frac{1}{k}$
Solving the expressions, we get:
$k (a^{2} + b^{2}) = c^{2}$ (equation 1).
$k (b^{2} + c^{2}) = a^{2}$ (equation 2).
$k (c^{2} + a^{2}) = b^{2}$ (equation 3).
On adding all the equations, we get–
$(a^{2} + b^{2} + c^{2}) = k (a^{2} + b^{2} + b^{2} + c^{2} + c^{2} + a^{2})$
⇒ $(a^{2} + b^{2} + c^{2}) = 2 k (a^{2} + b^{2} + c^{2})$
⇒ $k = \frac{1}{2}$
Hence, the correct answer is $\frac{1}{2}$ .

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Question : If a2+b2c2=b2+c2a2=c2+a2b2=1k, (k≠0), then k=?Option 1: 2Option 2: 1Option 3: 0Option 4: 12

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Question : If $\frac{a^{2} + b^{2}}{c^{2}} = \frac{b^{2} + c^{2}}{a^{2}} = \frac{c^{2} + a^{2}}{b^{2}} = \frac{1}{k}$ , $(k \neq 0)$ , then $k$ =?
Option 1: $2$
Option 2: $1$
Option 3: $0$
Option 4: $\frac{1}{2}$