Question

Find the polar of the point \[\left( {4, - 1} \right)\] with respect to the circle $2{x^2} + 2{y^2} = 11$

Answer 1

Hint: This problem comes under analytic geometry which is also known as coordinate geometry. The Cartesian coordinate system in polar is a two-dimensional in which each point is determined by a distance from a reference point. In this problem we are going to substitute the polar point to the equation of the circle before that we need to convert the equation of the circle in Cartesian form and complete the step-by-step solution.

Formula used: Polar of a given point
Equation of a polar of the circle ${x^2} + {y^2} + 2gx + 2fy + c = 0$ with respect to the pole \[({x_1},{y_1})\] is
\[x{x_1} + y{y_1} + g(x + {x_1}) + f({y_1} + {y_2}) + c = 0\]

Complete step-by-step answer:
First, let the polar of the point be \[({x_1},{y_1})\]
Given that the equation of the circle is $2{x^2} + 2{y^2} = 11$
Now, we convert the equation of the circle in Cartesian form which is mention in formula used
Since, line passes through origin \[\left( {0,0} \right)\] the equation becomes
$ \Rightarrow 2x{x_1} + 2y{y_1} = 11$
Now, Substituting the point \[\left( {4, - 1} \right)\] into the equation in the place of $({x_1},{y_1})$, we get
$ \Rightarrow 2x(4) + 2y( - 1) = 11$
Now, multiplying the numerical values and the variable remains same
$\therefore 8x - 2y = 11$ is the required polar.

Note: Thus we find the polar equation of the circle. This type of problem needs attention on analytic geometry on plane form and needs to know about Cartesian form and equations of circle. This may also need to think of basic algebraic calculation and the answer should be in the form of a line equation. We may also need to go through the question clearly because there might be slight changes to it.