First slide

Pair of straight lines

Question

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

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 ${ax}^{2} + 2 hxy + {by}^{2} = 0 .$ Then, $m + m^{2} = - \frac{2 h}{b}$
and ${mm}^{2} = \frac{a}{b}$ or $m^{3} = \frac{a}{b}$
Now, ${(m + m^{2})}^{3} = {(- \frac{2 h}{b})}^{3}$
$∴ m^{3} + m^{6} + 3 {mm}^{2} (m + m^{2}) = - \frac{8 h^{3}}{b^{3}}$
$∴ \frac{a}{b^{2}} (a + b) + \frac{8 h^{3}}{b^{3}} = \frac{6 ah}{b^{2}}$
$∴ \frac{a + b}{h} + \frac{8 h^{2}}{ab} = 6$
These are the set of parallel lines and the distance between parallel lines are equal. So, the figure is a rhombus.

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