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(B) Let the equation of a line passing through the origin with slope $m$ be $y - mx = 0$,or $mx - y = 0$. According to the given information,the perpe

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$7y^2 + 24xy = 0$

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(B) Let the equation of a line passing through the origin with slope $m$ be $y - mx = 0$,or $mx - y = 0$. According to the given information,the perpendicular distance from the point $(3, 4)$ to the line is $4$ units. Using the distance formula: $\frac{|m(3) - 1(4)|}{\sqrt{m^2 + (-1)^2}} = 4$. Squaring both sides: $\frac{(3m - 4)^2}{m^2 + 1} = 16$. $9m^2 - 24m + 16 = 16m^2 + 16$. $7m^2 + 24m = 0$. $m(7m + 24) = 0$,so $m = 0$ or $m = -\frac{24}{7}$. The equations of the lines are $y = 0$ and $y = -\frac{24}{7}x$,which is $7y + 24x = 0$. The combined equation is $y(7y + 24x) = 0$,which simplifies to $7y^2 + 24xy = 0$. Thus,option $B$ is correct.

If each line of a pair of lines passing through the origin is at a perpendicular distance of $4$ units from the point $(3, 4)$,then the equation of the pair of lines is

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