The following notation( $S_{1}$ ) in the circle is used to find the tangent to the circle at point $(x_{1}, y_{1})$
$S \equiv x^{2} + y^{2} + 2 g x + 2 f y + c$
$S_{1} \equiv x x_{1} + y y_{1} - g (x + x_{1}) - f (y + y_{1}) + c$

True

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False

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Solution

The correct option is B False
Actually tangent at point $(x_{1}, y_{1})$ to the circle is given by $T = 0$
$\Rightarrow x (x_{1}) + y (y_{1}) + g (x + x_{1}) + y (f + f_{1}) + c = 0$
$∴ B)$ False.

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Similar questions

The equation of tangent to the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ at its point $(x_{1}, y_{1})$ is given by $x x_{1} + y y_{1} + g (x + x_{1}) + f (y + y_{1}) + c = 0$

If $(x_{1}, y_{1})$ is a point inside the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ Given two expressions $S_{1}$ and $T_{1}$ such that,

$S_{1} = x_{1}^{2} + {y_{1}}^{2} + 2 g x_{1} + 2 f y_{1} + c$

$T_{1} = x x_{1} + y y_{1} + g (x + x_{1}) + f (y + y_{1}) + c$

Then the equation of chord centered at $(x_{1}, y_{1})$ is

A circle is of the form $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ and a pair of tangents are drawn from $(x_{1}, y_{1})$ to the circle.The combined equation of tangents is $S S_{1} = T^{2} .$

Where $S = x^{2} + y^{2} + 2 g x + 2 f y + c$

$S_{1} = x_{1}^{2} + y_{1}^{2} + 2 g x_{1} + 2 f y_{1} + c$

$T = x x_{1} + y y_{1} + g (x + x_{1}) + f (y + y_{1}) + c$

Q. Assertion :The tangent to the circle

x^{2} + y^{2} = 5

at the point

(1, - 2)

also touches the circle

x^{2} + y^{2} - 8 x + 6 y + 20 = 0

. Then its point of contact is

(3, - 1)

Reason: The equation of tangent to the circle

x^{2} + y^{2} + 2 g x + 2 f y + c = 0