Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade None
Question:
If $\frac{a}{\sqrt{bc}} - 2 = \sqrt[3]{\frac{b}{c}} + \sqrt[3]{\frac{c}{b}}$ where $a, b, c > 0$, then family of lines $\sqrt{ax} + \sqrt{by} + \sqrt{c} = 0$ passes through the point:
$(1, 1)$
$(1, -2)$
$(-1, 2)$
$(-1, 1)$
Step-by-Step Solution
Key Concept: Algebraic constraints on coefficients force a line to pass through specific fixed points.
Starting from $a - 2\sqrt{bc} = b + c$, rearrange to $(\sqrt{b}+\sqrt{c})^2 - (\sqrt{a})^2 = 0$, giving $\sqrt{b}+\sqrt{c}-\sqrt{a} = 0$ (rejected case leads to contradiction). The equation $\sqrt{ax} + \sqrt{by} + \sqrt{c} = 0$ passes through the fixed point $(-1, 1)$ because these coefficients satisfy the constraint $\sqrt{a} - \sqrt{b} - \sqrt{c} = 0$.
Correct Answer: 4