Find the pole of the straight line 2x - y = 1 with respect to the circle $5 x^{2} + 5 y^{2} = 9 .$

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Solution

Polar of the point $(x_{1}, y_{1})$ with respect to the circle $5 x^{2} + 5 y^{2} = 9$ is given by $5 x x_{1} + 5 y y_{1} = 9$
Comparing the above equation with the equation of the straight line $2 x - y = 1$ , we have
$\frac{5 x_{1}}{2} = \frac{5 y_{1}}{- 1} = \frac{9}{1}$
$\Rightarrow 5 x_{1} = 9 \times 2 = 18$
or $x_{1} = \frac{18}{5}$
Also, $5 y_{1} = - 9$ or $y_{1} = \frac{- 9}{5}$
The pole is thus $(\frac{18}{5}, \frac{- 9}{5})$

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