First slide

Pair of straight lines

Question

If the slope of one of the lines represented by the equation $a^{3} x^{2} - 2 h x y + b^{3} y^{2} = 0$ be the square of other then $a b^{2}, h, a^{2} b$ are in

Moderate

A

A.P
B

G.P
C

H.P
D

A.G.P

Solution

$m + m^{2} = \frac{2 h}{b^{3}} ......... (1)$ ; $m (m^{2}) = \frac{a^{3}}{b^{3}} \Rightarrow m = \frac{a}{b}$
from (1) $\Rightarrow m = \frac{a}{b} + \frac{a^{2}}{b^{2}} = \frac{2 h}{b^{3}} \Rightarrow a b^{2} + a^{2} b = 2 h$

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