If $\frac{1}{\sqrt{b} + \sqrt{c}}, \frac{1}{\sqrt{c} + \sqrt{a}}, \frac{1}{\sqrt{a} + \sqrt{b}}$ are in A.P., then the family of lines $a x + b y + c = 0$ will always passes through the fixed point:

(1, 1)

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(1, - 2)

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(- 1, 2)

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(1, 1)

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Solution

The correct option is B $(1, - 2)$
$\frac{1}{\sqrt{a} + \sqrt{c}} - \frac{1}{\sqrt{b} + \sqrt{c}} = \frac{1}{\sqrt{b} + \sqrt{a}} - \frac{1}{\sqrt{a} + \sqrt{c}} \Rightarrow \frac{\sqrt{b} - \sqrt{a}}{\sqrt{b} + \sqrt{c}} = \frac{\sqrt{c} - \sqrt{b}}{\sqrt{b} + \sqrt{a}} \Rightarrow a + c = 2 b$

So, $a, b, c$ are in A.P.
Hence, line $a x + b y + c = 0$ passes through $(1, - 2)$

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If 1√b+√c, 1√c+√a, 1√a+√b are in A.P., then the family of lines ax+by+c=0 will always passes through the fixed point:

The correct option is B (1,−2)1√a+√c−1√b+√c=1√b+√a−1√a+√c⇒√b−√a√b+√c=√c−√b√b+√a⇒a+c=2b So, a,b,c are in A.P. Hence, line ax+by+c=0 passes through (1,−2)

If $\frac{1}{\sqrt{b} + \sqrt{c}}, \frac{1}{\sqrt{c} + \sqrt{a}}, \frac{1}{\sqrt{a} + \sqrt{b}}$ are in A.P., then the family of lines $a x + b y + c = 0$ will always passes through the fixed point: