Find All Numbers x

**magentarita** · October 4th 2008, 03:07 AM

If P = (-3, 1) and Q = (x, 4), find all numbers x such that the vector represented by line PQ has length 5.

**earboth** · October 4th 2008, 04:10 AM

Originally Posted by magentarita

If P = (-3, 1) and Q = (x, 4), find all numbers x such that the vector represented by line PQ has length 5.

1. Calculate the vector $\overrightarrow{PQ} = \overrightarrow{OQ} - \overrightarrow{OP} = ((x-(-3)), 3)$

2. Calculate the magnitude of the vector $\overrightarrow{PQ}$ which has to be 5:

$\sqrt{(x+3)^2+3^2} = 5$ Square both sides:

$(x^2+6x+9) +9 = 25~\implies~x^2+6x-7=0~\implies~x=-7~\vee~x=1$

3. Therefore $Q_1 (-7,4)$ or $Q_2(1,4)$

**magentarita** · October 4th 2008, 04:55 AM

Originally Posted by earboth

1. Calculate the vector $\overrightarrow{PQ} = \overrightarrow{OQ} - \overrightarrow{OP} = ((x-(-3)), 3)$

2. Calculate the magnitude of the vector $\overrightarrow{PQ}$ which has to be 5:

$\sqrt{(x+3)^2+3^2} = 5$ Square both sides:

$(x^2+6x+9) +9 = 25~\implies~x^2+6x-7=0~\implies~x=-7~\vee~x=1$

3. Therefore $Q_1 (-7,4)$ or $Q_2(1,4)$

Another great job here.

Thanks

**bkarpuz** · October 4th 2008, 09:07 AM

Originally Posted by magentarita

If P = (-3, 1) and Q = (x, 4), find all numbers x such that the vector represented by line PQ has length 5.

Note that $Q\in\mathcal{C}$ , where $\mathcal{C}$ is the circle centered at $P$ with the radius $r=5$ .
Just write the equation of $\mathcal{C}$ and substitute $Q$ to this equation and solve $x$ as earboth done.

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