Question

The line passing through \[\left( -1,\dfrac{\pi }{2} \right)\] and perpendicular to \[\sqrt{3}\sin \theta +2\cos \theta =\dfrac{4}{r}\] is:
(a) \[2=\sqrt{3}r\cos \theta -2r\sin \theta \]
(b) \[5=-2\sqrt{3}r\sin \theta +4r\cos \theta \]
(c) \[2=\sqrt{3}r\cos \theta +2r\cos \theta \]
(d) \[5=2\sqrt{3}r\sin \theta +4r\cos \theta \]

Answer 1

Hint: In the above question, we have the equation of the line in parametric form and we have to write the equation of the line perpendicular to the given line, so we will use the concept that we will substitute \[\theta \] by \[\left( {{90}^{o}}+\theta \right)\] and we get the equation of the line in parametric form perpendicular to it. Also, we have given the point so we will easily find the constant term of the equation.

Complete step by step answer:
We have been given the equation of the line:
\[\sqrt{3}\sin \theta +2\cos \theta =\dfrac{4}{r}\]
By substituting the value of \[\theta \] by \[\left( {{90}^{o}}+\theta \right)\], we get the equation of the perpendicular line as follows:
\[\sqrt{3}\sin \left( {{90}^{o}}+\theta \right)+2\cos \left( {{90}^{o}}+\theta \right)=\dfrac{k}{r}\]
Also, we know that \[\sin \left( {{90}^{o}}+\theta \right)=\cos \theta \] and \[\cos \left( {{90}^{o}}+\theta \right)=-\sin \theta \]
By substituting these values in the equation, we get,
\[\sqrt{3}\cos \theta -2\sin \theta =\dfrac{k}{r}\]
\[\Rightarrow \sqrt{3}r\cos \theta -2r\sin \theta =k\]
Also, we have been given that the line passes through \[\left( -1,\dfrac{\pi }{2} \right)\] which means it satisfies the equation.
\[\Rightarrow \sqrt{3}\left( -1 \right)\cos \dfrac{\pi }{2}-2\times \left( -1 \right)\sin \left( \dfrac{\pi }{2} \right)=k\]
\[\Rightarrow 0+2=k\]
\[\Rightarrow k=2\]
So, we get the value of k as 2.
Hence, the equation of the line is perpendicular to the given lines and passes through \[\left( -1,\dfrac{\pi }{2} \right)\] is \[\sqrt{3}r\cos \theta -2r\sin \theta =2\]
Therefore, the correction option is (a).

Note: We can solve the given question by converting the equation from parametric to coordinate form. To change the parametric to coordinate form, we replace \[x\to r\cos \theta \] and \[y\to r\sin \theta \], also the given point is in the form of \[\left( r,\theta \right)\] we can also convert it into coordinates.