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Q.

if S=0 is the equation of pair of asymptotes of the hyperbola $3 x^{2} + 10 x y + 8 y^{2} + 14 x + 22 y + 7 = 0$ then the length of intercept
made by the pair of lines S=0 on x-axis is

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a

$\frac{2}{3}$

b

$\frac{4}{3}$

c

$1$

d

$\frac{1}{3}$

answer is C.

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Detailed Solution

$\begin{array}{l} 3 x^{2} + 10 x y + 8 y^{2} + 14 x + 22 y + c = 0 rep. Pair of lines \\ ∴ Δ = 0 \Rightarrow |\begin{array}{ccc} 3 & 5 & 7 \\ 5 & 8 & 11 \\ 7 & 11 & c \end{array}| = 0 \Rightarrow C = 1 \\ Put y = 0 \Rightarrow 3 x^{2} + 14 x + 15 = 0 \Rightarrow cuts x -axis at (\frac{- 9}{3}, \frac{- 5}{3}) \end{array}$

$∴ length intercept = \frac{4}{3}$

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