A straight line passes through the points of intersection $x - 2 y - 2 = 9$ and $2 x - b x - 6 = 0$ and the origin. Then the complete set of values ob $b$ for which the acute angle between this line and $y = 0$ is less $45^{o}$ is

(- \infty, 4) \cup (7, \infty)

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(- \infty, 5) \cup (7, \infty)

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(- \infty, 4) \cup (5, 7) \cup (7, \infty)

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(- \infty, 4) \cup (4, 5) \cup (7, \infty)

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Solution

The correct option is B $(- \infty, 5) \cup (7, \infty)$
$\begin{matrix} S o l v i n g t h e g i v e n e q u a t i o n x - 2 y - 2 = 0 2 y - 6 y - 6 = 0 - - - - - - - - - - - y (6 - 4) = 2 ∴ y = \frac{2}{4 - 6} P u t t i n g x w e g e t x = \frac{12 - 26}{4 - 6} e q u^{n} o f l i n e p a s s i n g t h r o u g h (0, 0) a n d (\frac{12 - 26}{4 - 6}, \frac{2}{67 - 6}) y - 0 = \frac{2}{12 - 16} (x - 0) y = \frac{1}{6 - 6} . x m = tan θ θ < 45^{\circ} m < 1 & m < 1 ∣ ∣ ∣ \frac{1}{6 - b} ∣ ∣ ∣ < 1 b < 5 - - - - (1) a n d b > 7 O n c o m b a i n g w e g e t (- \infty, 5) \cup (7, \infty) H e n c e, o p t i o n B i s t h e c o r r e c t a n s w e r . \end{matrix}$

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A straight line passes through the points of intersection x−2y−2=9 and 2x−bx−6=0 and the origin. Then the complete set of values ob b for which the acute angle between this line and y=0 is less 45o is

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