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Q.

The condition that one of the pair of lines $a x^{2} + 2 h x y + b y^{2} = 0$ be coincident with one line of the pair $3 x^{2} + 12 x y + 2 y^{2} = 0$ and the remaining lines are at right angles, then $h (a - b) =$

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a

$a + b$

b

$2 a b$

c

$\frac{a}{b}$

d

$a b$

answer is B.

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Detailed Solution

$F r o m t h e g i v e n d a t a r e q . c o n d i t i o n i s h_{1} (\frac{1}{a_{1}} - \frac{1}{b_{1}}) = h_{2} (\frac{1}{a_{2}} - \frac{1}{b_{2}}) w h e r e a_{1} = a, h_{1} = h, b_{1} = b a n d a_{2} = 3, h_{2} = 6, b_{2} = 2 \Rightarrow h (\frac{1}{a} - \frac{1}{b}) = 6 (\frac{1}{3} - \frac{1}{2}) \Rightarrow h (a - b) = 6 a b$

$\Rightarrow h (\frac{1}{a} - \frac{1}{b}) = 6 (\frac{1}{3} - \frac{1}{2})$

$\Rightarrow h (a - b) = a b$

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The condition that one of the pair of lines ax2+2hxy+by2=0 be coincident with one line of the pair 3x2+12xy+2y2=0 and the remaining lines are at right angles, then h(a−b)=