When is the line lx + my + n = 0 a tangent to | vedclass

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Question

(C) The given line is $lx + my + n = 0$.The line is tangent to the circle $x^2 + y^2 = r^2$ if and only if the perpendicular distance from the center

Vedclass · Accepted Answer

$n^2 = r^2 (l^2 + m^2)$

Vedclass · Answer

(C) The given line is $lx + my + n = 0$.The line is tangent to the circle $x^2 + y^2 = r^2$ if and only if the perpendicular distance from the center $(0, 0)$ to the line is equal to the radius $r$ of the circle.The perpendicular distance $d$ from $(0, 0)$ to $lx + my + n = 0$ is given by:$d = \frac{|l(0) + m(0) + n|}{\sqrt{l^2 + m^2}} = \frac{|n|}{\sqrt{l^2 + m^2}}$Setting $d = r$:$\frac{|n|}{\sqrt{l^2 + m^2}} = r$Squaring both sides:$\frac{n^2}{l^2 + m^2} = r^2$Therefore,$n^2 = r^2(l^2 + m^2)$.

When is the line $lx + my + n = 0$ a tangent to the circle $x^2 + y^2 = r^2$?

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