$x^{2} + y^{2} + 6 x + 8 y - 11 = 0$ ( chord of contact)

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Solution

For $x^{2} + y^{2} + 2 g x + 2 f x + c = 0$
Chord of contact from $(x_{1}, y_{1})$ is
$x x_{1} + y y_{1} + g (x + x_{1}) + f (y + y_{1}) + c = 0$
So, for $x^{2} + y^{2} + 6 x + 8 y - 11 = 0$
Chord of contact from $(x_{1}, y_{1})$ is
$x x_{1} + y y_{1} + 3 (x + x_{1}) + 4 (y + y_{1}) - 11 = 0$
1) For $(1, 1)$ , equation is
$x .1 + y .1 + 3 (x + 1) + 4 (y + 1) - 11 = 0 \Rightarrow 4 x + 5 y - 4 = 0$
2) For $(0, 0)$ , equation is
$x .0 + y .0 + 3 (x + 0) + 4 (y + 0) - 11 = 0 \Rightarrow 3 x + 4 y - 11 = 0$
3) For $(1, - 1)$ , equation is
$x .1 + y . - 1 + 3 (x + 1) + 4 (y - 1) - 11 = 0 \Rightarrow 4 x + 3 y - 12 = 0$
4) For $(- 1, - 1)$ , equation is
$x . - 1 + y . - 1 + 3 (x - 1) + 4 (y - 1) - 11 = 0 \Rightarrow 2 x + 3 y - 18 = 0$

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