Question

If the slope of one of the lines represented by ax² + 2hxy + by² = 0 be the square of the other, then $\frac{a + b}{h}$ + $\frac{8h^{2}}{ab}$=

• A. 4
• B. 6
• C. 8
• D. None of these

Answer 1

Answer: (B)

6

Answer 2

Let m and m² be the slopes of the lines represented by ax² + 2hxy + by² = 0

Then, m + m² = –$\frac{2h}{b}$ … (1)

m . m² = $\frac{a}{b}$ or m³ =$\frac{a}{b}$ … (2)

from (1) (m + m²)³ = $\left( - \frac{2h}{b} \right)^{3}$

Ž m³ + m⁶ + 3.m.m² (m + m²) = –$\frac{8h^{3}}{b^{3}}$

Ž $\frac{a}{b}$ + $\frac{a^{2}}{b^{2}}$ + $\frac{3a}{b}$ $\left( - \frac{2h}{b} \right)$ = $- \frac{8h^{3}}{b^{3}}$ {from (1) and (2)}

Ž $\frac{a}{b^{2}}$ (a + b) +$\frac{8h^{3}}{b^{3}}$ = $\frac{6ah}{b^{2}}$

or $\frac{(a + b)}{h}$ + $\frac{8h^{2}}{ab}$ = 6