Straight LineHardBloom L3
Question
If $\frac{a}{\sqrt{bc}} - 2 = \sqrt{\frac{b}{c}} + \sqrt{\frac{c}{b}}$ where $a,b,c > 0$, then family of lines $\sqrt{a}x + \sqrt{b}y + \sqrt{c} = 0$ passes through the point:
Options
A.$(1,1)$
B.$(1, - 2)$
C.$( - 1,2)$
D.$( - 1,1)$
Solution
Sol. [Key Concept: A family $ax + by + c = 0$ subject to a linear relation among $a,b,c$ passes through the fixed point read off that relation.]
$${a - 2\sqrt{bc} = b + c }{\Rightarrow (\sqrt{b} + \sqrt{c})^{2} - (\sqrt{a})^{2} = 0 }{\Rightarrow \ \sqrt{b} + \sqrt{c} - \sqrt{a} = 0 }$$or $\ \sqrt{b} + \sqrt{c} + \sqrt{a} = 0$ (rejected)
$$\sqrt{b} + \sqrt{c} - \sqrt{a} = 0$$
$\Rightarrow \sqrt{a}x + \sqrt{b}y + \sqrt{c} = 0$ passes through fixed point $( - 1,1)$.
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