Answer
Verified
439.5k+ views
Hint: We will locate the point on the graph. We will sketch the required line according to the conditions given in the question. We will convert the equation of the line into the polar form using the formulas for converting rectangular coordinates to polar coordinates.
Formulas used:
We will use following formulas for converting rectangular coordinates to polar coordinates:
\[x = r\cos \theta \]
\[y = r\sin \theta \]
\[\sqrt {{x^2} + {y^2}} = r\]
Complete step-by-step answer:
The point \[\left( {4,\dfrac{{2\pi }}{2}} \right)\] is equivalent to the point \[\left( {4,\pi } \right)\]. The modulus of the point represented by \[\left( {4,\dfrac{{2\pi }}{2}} \right)\] is 4 and it makes an angle of \[\pi \] radians with the positive \[x\] - axis. We will locate this point on the graph:
The point Z in red colour represents the required point. The point Z has value \[\left( {4,\pi } \right)\] in polar coordinates and value \[\left( { - 4,0} \right)\] in rectangular coordinates.
We can see from the figure that the line that is joining the point to the origin is the \[x\] - axis. We need a line that is perpendicular to the line joining the origin to this point i.e. our line will be perpendicular to the \[x\] - axis; this means that our line is parallel to the \[y\] - axis.
We need a line that passes through the point \[\left( { - 4,0} \right)\]and is parallel to the \[y\] - axis. We will represent this condition on the graph:
We can see from the graph that the required line is \[x = - 4\].
We will convert the notation of this line to polar coordinates. We will substitute \[ - 4\] for \[x\] in the 1st formula:
\[r\cos \theta = - 4\]
$\therefore $ The polar equation of the line passing through \[\left( {4,\dfrac{{2\pi }}{2}} \right)\] and perpendicular to the line joining the origin to this point is \[r\cos \theta = - 4\].
Note: We can also find the polar equation of the line using the formula \[r\cos \left( {\theta - \alpha } \right) = p\] where \[p\] is the length of the normal to the line from the origin and \[\alpha \] is the angle that the line makes with the polar axis. We will substitute 4 for \[p\] and \[\pi \] for \[\alpha \] in the formula for the polar equation of a line:
\[r\cos \left( {\theta - \pi } \right) = 4\].
We know that \[\cos \left( {\theta - \pi } \right) = - \cos \theta \]:
\[\begin{array}{c} \Rightarrow - r\cos \theta = 4\\ \Rightarrow r\cos \theta = - 4\end{array}\]
Formulas used:
We will use following formulas for converting rectangular coordinates to polar coordinates:
\[x = r\cos \theta \]
\[y = r\sin \theta \]
\[\sqrt {{x^2} + {y^2}} = r\]
Complete step-by-step answer:
The point \[\left( {4,\dfrac{{2\pi }}{2}} \right)\] is equivalent to the point \[\left( {4,\pi } \right)\]. The modulus of the point represented by \[\left( {4,\dfrac{{2\pi }}{2}} \right)\] is 4 and it makes an angle of \[\pi \] radians with the positive \[x\] - axis. We will locate this point on the graph:
The point Z in red colour represents the required point. The point Z has value \[\left( {4,\pi } \right)\] in polar coordinates and value \[\left( { - 4,0} \right)\] in rectangular coordinates.
We can see from the figure that the line that is joining the point to the origin is the \[x\] - axis. We need a line that is perpendicular to the line joining the origin to this point i.e. our line will be perpendicular to the \[x\] - axis; this means that our line is parallel to the \[y\] - axis.
We need a line that passes through the point \[\left( { - 4,0} \right)\]and is parallel to the \[y\] - axis. We will represent this condition on the graph:
We can see from the graph that the required line is \[x = - 4\].
We will convert the notation of this line to polar coordinates. We will substitute \[ - 4\] for \[x\] in the 1st formula:
\[r\cos \theta = - 4\]
$\therefore $ The polar equation of the line passing through \[\left( {4,\dfrac{{2\pi }}{2}} \right)\] and perpendicular to the line joining the origin to this point is \[r\cos \theta = - 4\].
Note: We can also find the polar equation of the line using the formula \[r\cos \left( {\theta - \alpha } \right) = p\] where \[p\] is the length of the normal to the line from the origin and \[\alpha \] is the angle that the line makes with the polar axis. We will substitute 4 for \[p\] and \[\pi \] for \[\alpha \] in the formula for the polar equation of a line:
\[r\cos \left( {\theta - \pi } \right) = 4\].
We know that \[\cos \left( {\theta - \pi } \right) = - \cos \theta \]:
\[\begin{array}{c} \Rightarrow - r\cos \theta = 4\\ \Rightarrow r\cos \theta = - 4\end{array}\]
Recently Updated Pages
what is the correct chronological order of the following class 10 social science CBSE
Which of the following was not the actual cause for class 10 social science CBSE
Which of the following statements is not correct A class 10 social science CBSE
Which of the following leaders was not present in the class 10 social science CBSE
Garampani Sanctuary is located at A Diphu Assam B Gangtok class 10 social science CBSE
Which one of the following places is not covered by class 10 social science CBSE
Trending doubts
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The only snake that builds a nest is a Krait b King class 11 biology CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Why is there a time difference of about 5 hours between class 10 social science CBSE
Which places in India experience sunrise first and class 9 social science CBSE