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 January 24th, 2012, 08:48 PM #1 Newbie   Joined: Jan 2012 Posts: 21 Thanks: 0 straight line problem 1/r+1/m=1/c r,m are variables c is a constant show that x/r+y/m=1 is going through a constant point
 January 24th, 2012, 09:21 PM #2 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: straight line problem We are given the family of lines: $\frac{x}{r}+\frac{y}{m}=1$ where $\frac{1}{r}+\frac{1}{m}=\frac{1}{c}$ with c a non-zero constant. If this family of lines all pass through the same point, let's call this point (h,k). We should then be able to express the family of lines in the form: $y-k=m_f(x-h)$ where $m_f\in\mathbb R$ is an arbitrary slope. $\frac{m+r}{mr}=\frac{1}{c}$ $mr=c(m+r)$ $mr-cr=cm$ $r(m-c)=cm$ $r=\frac{cm}{m-c}$ $\frac{x}{r}+\frac{y}{m}=1$ $\frac{mx+ry}{mr}=1$ $mx+ry=mr=cm+cr$ $r(y-c)=m(c-x)$ $\frac{cm}{m-c}(y-c)=m(c-x)$ $\frac{c}{m-c}(y-c)=(c-x)$ $c(y-c)=(c-x)(m-c)$ $y-c=\frac{c-m}{c}(x-c)$ So we find all the lines pass through the point (c,c).
 January 24th, 2012, 10:46 PM #3 Newbie   Joined: Jan 2012 Posts: 21 Thanks: 0 Re: straight line problem thanks a lot...this site seems good. :P
 January 24th, 2012, 10:48 PM #4 Senior Member     Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,211 Thanks: 521 Math Focus: Calculus/ODEs Re: straight line problem Glad to help, and welcome to the forum! I'm glad you like this forum!

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