[Math] Equation of a line passing through the origin

geometry

I have the question "Write down the equation of the line passing through the origin and perpendicular to y = 2X + 3".

I know that the equation of a line is y = mx + c and that the origin is (0,0). However, I am not sure how I would start this.

Best Answer

An important fact about lines is that they are perpendicular if and only if the slope of one is the opposite reciprocal of the slope of the other. That is, $y=mx+b$ and $y=nx+c$ are perpendicular if and only if $m=-\frac{1}{n}$ (equivalently, $n=-\frac{1}{m}$.

You want your line to be perpendicular to $y=2x+3$, so your line must have slope $m=-\frac{1}{2}$.

Then you have an equation of the form $y=-\frac{1}{2}x+c$, and you need to solve for $c$. To do this, use the fact that your line goes through $(0,0)$. If you plug in $x=0$ and $y=0$ to the equation of the line, you can solve for $c$.

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