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The line lx +...

Question

The line $l x + m y + n = 0$ is normal to the circle $x^{2} + y^{2} + 2 g x + 2 f y + c = 0,$ if

A

l g + m f + n = 0

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B

l g + m f - n = 0

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C

l g - m f - n = 0

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D

l g - m f + n = 0

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Solution

The correct option is B $l g + m f - n = 0$
Given circle is
$x^{2} + y^{2} + 2 g x + 2 f y + c = 0$
Centre, $C = (- g, - f)$
Line $l x + m y + n = 0$ is normal to the given circle, so it passes through the centre,
$∴ - l g - m f + n = 0$
$\Rightarrow l g + m f - n = 0$

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Similar questions

Q. The condition for the line

l x + m y + n = 0

to be a normal to the circle

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

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

intercepts on the line lx+ my = 1, a length which subtends a right angle at origin then

\frac{l g + m f + 1}{l^{2} + m^{2}} =

l x + m y + n = 0

is normal to the circle

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

l x + m y + n = 0

touches a fixed circle

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

4 l^{2} + 3 m^{2} + n^{2} + 2 m n = 0

| g + f + c |

l x + m y + n = 0

is a tangent line to the circle

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

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