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The line $a x+b y+c=0$ is normal to the circle $x^2+y^2+2 g x+2 f y+d=0$ if
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Verified Answer
The correct answer is:
$a g+b f-c=0$
Given, equation of circle is
$$
x^2+y^2+2 g x+2 f y+d=0
$$
Its centre will be $(-g,-f)$.
A normal to this circle will always passes through centre
Given, equation of line is $a x+b y+c=0$
If $a x+b y+c=0$, then it satisfied $(-g,-f)$
$$
\begin{aligned}
& \Rightarrow a(-g)+b(-f)+c & =0 \\
\Rightarrow & a g+b f-c & =0
\end{aligned}
$$
$$
x^2+y^2+2 g x+2 f y+d=0
$$
Its centre will be $(-g,-f)$.
A normal to this circle will always passes through centre
Given, equation of line is $a x+b y+c=0$
If $a x+b y+c=0$, then it satisfied $(-g,-f)$
$$
\begin{aligned}
& \Rightarrow a(-g)+b(-f)+c & =0 \\
\Rightarrow & a g+b f-c & =0
\end{aligned}
$$
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