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Straight line

Posted November 21st, 2009 by alpha
in

If the equations (q-r)x+(r-p)y+(p-q)=0
and$ (q^3-r^3)x+(r^3-p^3)y+(p^3-q^3)=0 $represent the same line , then prove that either p=q or q=r or p+q+r=0

‹ Straight line Coordinate Geometry ›

Comments

My solution:

On November 21st, 2009 alpha says:

If the equations represent the same line, their slopes must be equal.
slope of eqn.1=slope of eqn.2
$ \frac{r-q}{r-p}=\frac{r^3-q^3}{r^3-p^3} $
on solving further I got p+q+r=0

Can I say that the other two conditions can be proved by simple observation?
(Will it be sufficient for proving?)

-------------------x----------------------------
``Old theorems never die; they turn into definitions.''
----E. Hewitt

''ALPHA" =)

answer

On November 21st, 2009 Structure says:

Lines

$$ax+by+c=0$$

and

$$Ax+By+C=0$$

are identical if and only if

$$\frac{a}{A}=\frac{b}{B}=\frac{c}{C}$$

with convention that if A=0 then a=0 and so on.(B=0 than b=0 and C=0 than c=0)

$$\frac{a}{A}=\frac{b}{B}$$

is not enough, they are just parallel.

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