How does equation of a line change as scale of axes changes?

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I know a way to find new equation of line but it's a bit lengthy. I can first get any two different points on line, then scale those points according to the scale change of the axes, and finally go back to line.



But I was wondering how can I directly get a new line equation from the original one. For instance, if I have a line of form $ax+by+c=0$, how I can get new parameters $[a, b, c]$ for this line when the x and y axes have changed by scales $alpha$ and $beta$. Thanks.




geometry




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    $frac aalpha x + frac bbeta y + c = 0$.
    – fleablood
    Jul 27 at 15:22















up vote
1
down vote

favorite












I know a way to find new equation of line but it's a bit lengthy. I can first get any two different points on line, then scale those points according to the scale change of the axes, and finally go back to line.



But I was wondering how can I directly get a new line equation from the original one. For instance, if I have a line of form $ax+by+c=0$, how I can get new parameters $[a, b, c]$ for this line when the x and y axes have changed by scales $alpha$ and $beta$. Thanks.




geometry




share|cite|improve this question















  • 1




    $frac aalpha x + frac bbeta y + c = 0$.
    – fleablood
    Jul 27 at 15:22













up vote
1
down vote

favorite









up vote
1
down vote

favorite











I know a way to find new equation of line but it's a bit lengthy. I can first get any two different points on line, then scale those points according to the scale change of the axes, and finally go back to line.



But I was wondering how can I directly get a new line equation from the original one. For instance, if I have a line of form $ax+by+c=0$, how I can get new parameters $[a, b, c]$ for this line when the x and y axes have changed by scales $alpha$ and $beta$. Thanks.




geometry




share|cite|improve this question











I know a way to find new equation of line but it's a bit lengthy. I can first get any two different points on line, then scale those points according to the scale change of the axes, and finally go back to line.



But I was wondering how can I directly get a new line equation from the original one. For instance, if I have a line of form $ax+by+c=0$, how I can get new parameters $[a, b, c]$ for this line when the x and y axes have changed by scales $alpha$ and $beta$. Thanks.





geometry





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share|cite|improve this question




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asked Jul 27 at 15:18









Newbie

184




184







  • 1




    $frac aalpha x + frac bbeta y + c = 0$.
    – fleablood
    Jul 27 at 15:22













  • 1




    $frac aalpha x + frac bbeta y + c = 0$.
    – fleablood
    Jul 27 at 15:22








1




1




$frac aalpha x + frac bbeta y + c = 0$.
– fleablood
Jul 27 at 15:22





$frac aalpha x + frac bbeta y + c = 0$.
– fleablood
Jul 27 at 15:22











1 Answer
1






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0
down vote



accepted










By the new coordinates $X$ and $Y$, we have



  • $X=alpha x$

  • $Y=beta y$

$$ax+by+c=0 implies frac a alphaX+frac b betaY+c=0 implies abeta X+balpha Y+alpha beta c=0$$






share|cite|improve this answer





















  • I tested and it works, thanks. I was multiplying instead of dividing :|
    – Newbie
    Jul 27 at 15:47






  • 1




    @Newbie ah ok! Indeed it is not so difficult. To avoid mistake we need to give a different name to the new coordinates. Bye
    – gimusi
    Jul 27 at 15:48










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1 Answer
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active

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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes








up vote
0
down vote



accepted










By the new coordinates $X$ and $Y$, we have



  • $X=alpha x$

  • $Y=beta y$

$$ax+by+c=0 implies frac a alphaX+frac b betaY+c=0 implies abeta X+balpha Y+alpha beta c=0$$






share|cite|improve this answer





















  • I tested and it works, thanks. I was multiplying instead of dividing :|
    – Newbie
    Jul 27 at 15:47






  • 1




    @Newbie ah ok! Indeed it is not so difficult. To avoid mistake we need to give a different name to the new coordinates. Bye
    – gimusi
    Jul 27 at 15:48














up vote
0
down vote



accepted










By the new coordinates $X$ and $Y$, we have



  • $X=alpha x$

  • $Y=beta y$

$$ax+by+c=0 implies frac a alphaX+frac b betaY+c=0 implies abeta X+balpha Y+alpha beta c=0$$






share|cite|improve this answer





















  • I tested and it works, thanks. I was multiplying instead of dividing :|
    – Newbie
    Jul 27 at 15:47






  • 1




    @Newbie ah ok! Indeed it is not so difficult. To avoid mistake we need to give a different name to the new coordinates. Bye
    – gimusi
    Jul 27 at 15:48












up vote
0
down vote



accepted







up vote
0
down vote



accepted






By the new coordinates $X$ and $Y$, we have



  • $X=alpha x$

  • $Y=beta y$

$$ax+by+c=0 implies frac a alphaX+frac b betaY+c=0 implies abeta X+balpha Y+alpha beta c=0$$






share|cite|improve this answer













By the new coordinates $X$ and $Y$, we have



  • $X=alpha x$

  • $Y=beta y$

$$ax+by+c=0 implies frac a alphaX+frac b betaY+c=0 implies abeta X+balpha Y+alpha beta c=0$$







share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











answered Jul 27 at 15:22









gimusi

64.9k73483




64.9k73483











  • I tested and it works, thanks. I was multiplying instead of dividing :|
    – Newbie
    Jul 27 at 15:47






  • 1




    @Newbie ah ok! Indeed it is not so difficult. To avoid mistake we need to give a different name to the new coordinates. Bye
    – gimusi
    Jul 27 at 15:48
















  • I tested and it works, thanks. I was multiplying instead of dividing :|
    – Newbie
    Jul 27 at 15:47






  • 1




    @Newbie ah ok! Indeed it is not so difficult. To avoid mistake we need to give a different name to the new coordinates. Bye
    – gimusi
    Jul 27 at 15:48















I tested and it works, thanks. I was multiplying instead of dividing :|
– Newbie
Jul 27 at 15:47




I tested and it works, thanks. I was multiplying instead of dividing :|
– Newbie
Jul 27 at 15:47




1




1




@Newbie ah ok! Indeed it is not so difficult. To avoid mistake we need to give a different name to the new coordinates. Bye
– gimusi
Jul 27 at 15:48




@Newbie ah ok! Indeed it is not so difficult. To avoid mistake we need to give a different name to the new coordinates. Bye
– gimusi
Jul 27 at 15:48












 

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