Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
If the slope of one of the line represented by $ax^2 + 2hxy + by^2 = 0$ is the square of the other, then the value of $\frac{a + b}{h} - \frac{8h^2}{ab}$ is ____.
Step-by-Step Solution
Key Concept: Relating multiple slope conditions requires systematic expansion and substitution of cubic and quadratic expressions.
Given $m + m^2 = -\frac{2h}{b}$ and $m^3 = \frac{a}{b}$, cube the first equation: $(m + m^2)^3 = -\frac{8h^3}{b^3}$. Expanding and using $m^3 + m^6 + 3m^2(m+m^2) = -\frac{8h^3}{b^3}$, substitute $m^3 = \frac{a}{b}$ and simplify to get $\frac{a(a+b)}{b^2} + \frac{8h^3}{b^3} - \frac{6ah}{b^2} = \frac{8h^3}{b^3}$, which yields $\frac{a+b}{h} - \frac{ab}{ab} = 6$.
Correct Answer: 6