Question

Find the polar of the point $\left( -2,3 \right)$ with respect to the circle ${{x}^{2}}+{{y}^{2}}-4x-6y+5=0$ .

Answer 1

Hint**:** We should know the general form of polar equation with respect to any point let say $\left( {{x}_{1}},{{y}_{1}} \right)$ which is given as $x{{x}_{1}}+y{{y}_{1}}+g\left( x+{{x}_{1}} \right)+f\left( y+{{y}_{1}} \right)+c=0$ . Then we will find the value of g and f by comparing the equation of circle given to us with general form ${{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0$ . After finding values of g, f and $\left( -2,3 \right)$ point we will substitute all the values in the polar equation, and we will get the answer.

Complete step-by-step answer :

Here, we should know the general form of polar equation with respect to any point let say $\left( {{x}_{1}},{{y}_{1}} \right)$ which is given as

$x{{x}_{1}}+y{{y}_{1}}+g\left( x+{{x}_{1}} \right)+f\left( y+{{y}_{1}} \right)+c=0$ …………………(1)

The general form of equation of circle is given as

${{x}^{2}}+{{y}^{2}}+2gx+2fy+c=0$

C is constant which we can directly write from the equation of circle given to us.

By comparing this with our equation of circle, we will get the value of g and f. So, we get on comparing as

$2gx=-4x$

On further solving and cancelling x term from both the sides, we can write it as

$g=-2$

Similarly, we get value of ‘f’ as

$2fy=-6y$

On cancelling a term from both sides and solving, we get value as

$f=-3$

Now, we will substitute values of g, f and point $\left( {{x}_{1}},{{y}_{1}} \right)$ which is given to us as $\left( -2,3 \right)$ in equation (1), we get as

$-2x+3y-2\left( x-2 \right)-3\left( y+3 \right)+5=0$

On further simplification, we get as

$-2x+3y-2x+4-3y-9+5=0$

We will cancel 3y terms and further solving, we get equation as

$-2x-2x+4-4=0$

$-4x=0$

Thus, on dividing both sides by $-4$ , we get

$x=0$

Thus, $x=0$ is the polar equation of the point $\left( -2,3 \right)$ with respect to the circle ${{x}^{2}}+{{y}^{2}}-4x-6y+5=0$ .

Note**:** Students sometimes do not understand what is asked in question and in order to just solve the problem put the given points in the equation of circle given to us i.e. ${{x}^{2}}+{{y}^{2}}-4x-6y+5=0$ . On solving this, we will get incorrect answers. So, first remember the general form of the polar equation and then solve this type of problem.