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Question: Angle between the lines \(3 x + y + 4 = 0\) is...

Angle between the lines 3x+y+4=03 x + y + 4 = 0 is

A

90∘90 ^ { \circ }

B

45∘45 ^ { \circ }

C

180∘180 ^ { \circ }

D

60∘60 ^ { \circ }

Answer

45∘45 ^ { \circ }

Explanation

Solution

tan⁡θ=∣a2b1−a1b2a1a2+b1b2∣=∣(3)(−1)−(2)(1)(3)(2)+(−1)(1)∣\tan \theta = \left| \frac { a _ { 2 } b _ { 1 } - a _ { 1 } b _ { 2 } } { a _ { 1 } a _ { 2 } + b _ { 1 } b _ { 2 } } \right| = \left| \frac { ( 3 ) ( - 1 ) - ( 2 ) ( 1 ) } { ( 3 ) ( 2 ) + ( - 1 ) ( 1 ) } \right|

⇒ tan⁡θ=∣−3−26−1∣=∣−55∣=∣−1∣\tan \theta = \left| \frac { - 3 - 2 } { 6 - 1 } \right| = \left| \frac { - 5 } { 5 } \right| = | - 1 |

θ=tan⁡−1∣−1∣=tan⁡−11=45∘\theta = \tan ^ { - 1 } | - 1 | = \tan ^ { - 1 } 1 = 45 ^ { \circ } .