First slide

Pair of straight lines

Question

If one of the lines given by the equation $2 x^{2} + a x y + 3 y^{2} = 0$ coincide with one of those given by $2 x^{2} + b x y - 3 y^{2} = 0$ and the other lines represented by them be perpendicular, then

Moderate

A

a = – 5, b = 1
B

a = 5, b = – 1
C

a = 5, b = 1
D

none of these

Solution

Let $\frac{2}{3} x^{2} + \frac{a}{3} x y + y^{2} = (y - m x) (y - m^{'} x)$
and $\frac{2}{- 3} x^{2} + \frac{b}{- 3} x y + y^{2} = (y + \frac{1}{m} x) (y - m^{'} x)$
then $m + m^{'} = - \frac{a}{3}, m m^{'} = \frac{2}{3}$ (i)
$\begin{array}{l} \frac{1}{m} - m^{'} = \frac{- b}{3}, - \frac{m^{'}}{m} = - \frac{2}{3} \\ \Rightarrow m^{2} = 1 \Rightarrow m = \pm 1 \end{array}$
If $\begin{array}{l} m = 1, m^{'} = \frac{2}{3} \Rightarrow a = - 5, b = - 1 \end{array}$
If $m = - 1, m^{'} = - \frac{2}{3} \Rightarrow a = 5, b = 1 .$

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