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[SOLVED] ratio of lengths

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Poirot

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Feb 15, 2012
250
Let B=(5,21), C=(40,-14) and $\frac{BP}{PC}$=1/6. What is the most efficent way to find P? My book just states P=(6/7)B+(1/7)C but no idea how they got that.
 

earboth

Active member
Jan 30, 2012
74
Let B=(5,21), C=(40,-14) and $\frac{BP}{PC}$=1/6. What is the most efficent way to find P? My book just states P=(6/7)B+(1/7)C but no idea how they got that.
Draw a sketch.
 

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chisigma

Well-known member
Feb 13, 2012
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Let B=(5,21), C=(40,-14) and $\frac{BP}{PC}$=1/6. What is the most efficent way to find P? My book just states P=(6/7)B+(1/7)C but no idea how they got that.
If x and y are the coordinates of P, then it must be...

$\displaystyle f(x,y)= 36\ (x-5)^{2} + 36\ (y-21)^{2} - (x-40)^{2} - (y+14)^{2}=0$ (1)

The (1) is a 'quadratic curve' as illustrated in...

Quadratic Curve -- from Wolfram MathWorld

Kind regards

$\chi$ $\sigma$
 

chisigma

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Feb 13, 2012
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Last edited:

soroban

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Feb 2, 2012
409
Hello, Poirot!

Did you make a sketch?


Let $B=(5,21),\;C=(40,\text{-}14),\;\dfrac{BP}{PC}=\dfrac{1}{6}$

What is the most efficent way to find $P$ ?

Code:
      |
      | (5,21)   +35
      |  B♥ → → → → → → +
      |     *           ↓
      |      Po         ↓
      |         *       ↓ -35
  ----+-----------*-----↓------
      |             *   ↓
      |               * ↓
      |                 ♥C
      |              (40,-14)
      |
Going from $B$ to $C$, we move 35 right and 35 down.

Point $P$ is $\tfrac{1}{7}$ of the way from $B$ to $C$.

The x-coordinate is $\tfrac{1}{7}$ of the way from $5$ to $40$.
. . Hence: .$x \;=\;5 + \tfrac{1}{7}(40-5) \;=\;5 + \tfrac{1}{7}(35) \;=\;5 + 5 \;=\;10$

The y-coordinate is $\tfrac{1}{7}$ of the way from $21$ to $\text{-}14.$
. . Hence: .$y \;=\;21 + \tfrac{1}{7}(\text{-}14 - 21) \;=\;21 + \tfrac{1}{7}(\text{-}35) \;=\;21 - 5 \;=\;16$


Therefore, $P$ is at $(10,16).$
 

CaptainBlack

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Jan 26, 2012
890
Let B=(5,21), C=(40,-14) and $\frac{BP}{PC}$=1/6. What is the most efficent way to find P? My book just states P=(6/7)B+(1/7)C but no idea how they got that.
If your book gives that answer then you have not posted the question as asked, or omitted implied side conditions for the question set.

The quoted answer implies that you want a point P between B and C, while the statement does not so constrain P and defines a locus in the plane.

In future please post the question as asked and include any additional constraints implied by the topic you are studying and or the common conditions in force on a question set.

CB
 
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Poirot

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Feb 15, 2012
250
I've solved this now thanks to the answers.
 

earboth

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Jan 30, 2012
74