If a line L passing through a point A(2,3) intersects another line 4x−3y−19=0 at the point B such that AB=4, then the angle made by the line L with positive X-axis in anticlockwise direction is
Let the angle made by Line L with positive X-axis be θ. So, B=(Ax+AB‌cos‌θ,Ay+ABsin‌θ) Given, A(2,3) and AB=4, we have B=(2+4‌cos‌θ,3+4sin‌θ) Since, point B lies on the line 4x−3y−19=0, substitute the coordinate of B into this equation. ‌4(2+4‌cos‌θ)−3(3+4sin‌θ)−19=0 ‌⇒‌‌8+16‌cos‌θ−9−12sin‌θ−19=0 ‌⇒‌‌16‌cos‌θ−12sin‌θ−20=0 ‌⇒‌‌4‌cos‌θ−3sin‌θ−5=0 ‌⇒‌‌‌
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‌cos‌θ−‌
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sin‌θ=1 Let cos‌α=‌
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and sin‌α=‌
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for some angle α in the first quadrant. ‌∴cos‌α⋅cos‌θ−sin‌α⋅sin‌θ=1 ‌⇒cos(α+θ)=1 ‌⇒α+θ=2nπ,‌ for some integer ‌n. ‌⇒α+θ=0 ‌⇒α=−θ‌ or ‌α+θ=2π ‌⇒θ=2π−α The angle made by the line L with positive X-axis in the anti-clockwise direction is taken as a positive angle, where cos‌θ=‌
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‌ and ‌sin‌θ=−‌
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This angle lies in the fourth quadrant. So, tan‌θ=‌