Question

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 $ax+by+c=0$ will always passes through the fixed point:

• A. $(1,1)$
• B. $(1,-2)$
• C. $(-1,2)$
• D. $(1,1)$

Answer 1

The correct option is B $(1,-2)$

$\frac{1}{\sqrt{a}+\sqrt{c}}-\frac{1}{\sqrt{b}+\sqrt{c}}=\frac{1}{\sqrt{b}+\sqrt{a}}-\frac{1}{\sqrt{a}+\sqrt{c}}\Rightarrow \frac{\sqrt{b}-\sqrt{a}}{\sqrt{b}+\sqrt{c}}=\frac{\sqrt{c}-\sqrt{b}}{\sqrt{b}+\sqrt{a}}\Rightarrow a+c=2b$

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