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Question: A person considers lines on the plane $\mathbb{R^2}$ to be solutions of equations of the form $ax+by+c=0$, where $a,$ $b,$ and $c$ are fixed reals satisfying $a^2+b^2\neq0$. Give a point $P=(x_0,y_0)$ show an equation of a line passing through $P$ and parallel to the line given by $ax+by+c=0$.

My work so far:

Lines that are parallel have the same slope. So, if I put $ax+by+c=0$ into slope intercept form, I end up with $y=\frac{-ax}{b}-\frac{c}{b}$. So my slope is $m=\frac{-a}{b}$. Using this information, I will find the y-intercept using the point $P$. So I get: $b=y_0-(\frac{-a}{b})x_0$ $\Rightarrow$ $b=y_0+\frac{a}{b}x_0$. Now, putting all this information together, back into the slope intercept form I have: $y=\frac{-a}{b}x+y_0-(\frac{-a}{b})x_0$.

I'm not sure if any of this is correct or not. Any help would be appreciated.

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  • $\begingroup$ Do you know $b \neq 0 $? maybe you need to consider the $b=0$ case separately. $\endgroup$ – JC574 Aug 7 '14 at 19:13
  • $\begingroup$ For an easier approach, note that the lines parallel to $ax+by+c=0$ have equations of the shape $ax+by+k=0$. $\endgroup$ – André Nicolas Aug 7 '14 at 19:14
  • $\begingroup$ How would you go about showing that with those two equations? $\endgroup$ – Michelle Aug 7 '14 at 19:16
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    $\begingroup$ Plug in $x=x_0$, $y=y_0$ and thus find $k$. $\endgroup$ – André Nicolas Aug 7 '14 at 19:19
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All lines parallel to $ax+by+c=0$ are of the form $ax+by+d=0$ for some $d$.

If the parallel line passes through $(x_0, y_0)$, then $ax_0+by_0+d=0$ or $d=-ax_0-by_0$.

The line is therefore $ax+by-ax_0-by_0=0$.

Note: I know this is substantially equivalent to some of the other answers, but I wanted to express it more simply.

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  • $\begingroup$ Would you do anything with the other information provided in the problem? This is where I am confused, we have been given the information of $a^2+b^2\neq0$ and I'm not sure if we need to use this or not in our solution? $\endgroup$ – Michelle Aug 7 '14 at 20:44
  • $\begingroup$ If both $a=0$ and $b=0$, then the equation $ax+by+c=0$ reduces to $c=0$, which is not the equation of a line. If $c=0$, then it's the whole plane, because any point $(x,y)$ satisfies it. If $c\neq0$, then it's an empty set of points. Including $a^2 + b^2 \neq 0$ in the question just rules out the case where both are $0$, so that all you're left with are valid linear equations. $\endgroup$ – Zimul8r Aug 7 '14 at 21:06

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