First slide

Pair of straight lines

Question

$If the slope of one of the lines represented by a x^{2} + 2 h x y + b y^{2} = 0 is the square of the other, then \frac{a + b}{h} + \frac{8 h^{2}}{a b} =$

Moderate

A

4
B

6
C

8
D

None of these

Solution

$Let m and m^{2} be the slopes of the lines represented by a x^{2} + 2 h x y + b y^{2} = 0 . Then,$
$\begin{array}{ll} m + m^{2} = - \frac{2 h}{b} \\ and m m^{2} = \frac{a}{b} or m^{3} = \frac{a}{b} \\ ∴ {(m + m^{2})}^{3} = {(- \frac{2 h}{b})}^{3} \\ ∴ m^{3} + m^{6} + 3 m m^{2} (m + m^{2}) = - \frac{8 h^{3}}{b^{3}} \\ \frac{a}{b} + {(\frac{a}{b})}^{2} + \frac{3 a}{b} (\frac{- 2 h}{b}) = - \frac{8 h^{3}}{b^{3}} \\ ∴ \frac{a}{b^{2}} (a + b) + \frac{8 h^{3}}{b^{3}} = \frac{6 a h}{b^{2}} \\ M u l t i p l y b o t h s i d e s b y \frac{b^{2}}{a h} \end{array}$
$∴ \frac{a + b}{h} + \frac{8 h^{2}}{a b} = 6$

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