Question

find the equation of the straight line upon which the length of perpendicular from the origin is 3root 2 units and this perpendicular makes an angel of 75 degree with the positive direction of x-axis?
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Answer 1

$\textbf{Given:}$

$\textsf{Length of the perpendicular from origin to the line is}\;\mathsf{\dfrac{\sqrt{3}}{2}}$

$\textsf{and the perrpendicular makes an angle of 75 degrees with x axis}$

$\textbf{To find:}$

$\textsf{Equation of the straight line}$

$\textbf{Solution:}$

$\textbf{Normal form of straight line:}$

$\boxed{\mathsf{x\;cos\alpha+y\;sin\alpha=p}}$

$\mathsf{Here,\;\alpha=75^\circ\;\&\;p=\dfrac{\sqrt{3}}{2}}$

$\textsf{The required line is}$

$\mathsf{x\;cos\alpha+y\;sin\alpha=p}$

$\mathsf{x\;cos75^\circ+y\;sin75^\circ=\dfrac{\sqrt{3}}{2}}$........(1)

$\mathsf{cos75^\circ=cos(30^\circ+45^\circ)}$

$\mathsf{cos75^\circ=cos30^\circ\,cos45^\circ-sin30^\circ\,sin45^\circ}$

$\mathsf{cos75^\circ=\dfrac{\sqrt{3}}{2}\,\dfrac{1}{\sqrt{2}}-\dfrac{1}{2}\,\dfrac{1}{\sqrt{2}}}$

$\implies\boxed{\mathsf{cos75^\circ=\dfrac{\sqrt{3}-1}{2\sqrt{2}}}}$

$\mathsf{sin75^\circ=sin(30^\circ+45^\circ)}$

$\mathsf{sin75^\circ=sin30^\circ\,cos45^\circ+cos30^\circ\,sin45^\circ}$

$\mathsf{cos75^\circ=\dfrac{1}{2}\,\dfrac{1}{\sqrt{2}}+\dfrac{\sqrt{3}}{2}\,\dfrac{1}{\sqrt{2}}}$

$\implies\boxed{\mathsf{sin75^\circ=\dfrac{\sqrt{3}+1}{2\sqrt{2}}}}$

$\mathsf{(1)\;becomes}$

$\mathsf{x\left(\dfrac{\sqrt{3}-1}{2\sqrt{2}}\right)+y\left(\dfrac{\sqrt{3}+1}{2\sqrt{2}}\right)=\dfrac{\sqrt{3}}{2}}$

$\mathsf{x\left(\dfrac{\sqrt{3}-1}{\sqrt{2}}\right)+y\left(\dfrac{\sqrt{3}+1}{\sqrt{2}}\right)=\sqrt{3}}$

$\mathsf{x(\sqrt{3}-1)+y(\sqrt{3}+1)=\sqrt{2}\sqrt{3}}$

$\mathsf{x(\sqrt{3}-1)+y(\sqrt{3}+1)=\sqrt{6}}$

$\implies\boxed{\mathsf{x(\sqrt{3}-1)+y(\sqrt{3}+1)-\sqrt{6}=0}}$

$\textbf{Find more:}$

In what follows, p denotes the distance of the straight line from the origin and a denotes the anglemade by the normal ray drawn from the origin to the straight line with Ox measured in the

anti-clockwise sense. Find the equations of the straight lines with the following values of

(1) p = 5, alpha = 60°

​https://brainly.in/question/20974481