Finding coordinates

1

12345678

Junior Member
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Mar 30, 2013
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‘The line L has gradient -2 and passes through the point A (3,5). B is a point of the line L such that the distance AB is 6√5. Find thecoordinates of each of the possible points B’.
So I first calculated the equation of L…
Y - Y₁ = m (X - X₁)
Y – 5 = -2 (X – 3)
Y – 5 = -2X + 6
Y = 11 – 2X
So I then drew a triangle to try and visualise the problem---and I got …
(X – 3)² + (Y – 5)² = (6√5)²
Giving … X² + Y² - 6X – 10Y = 146
However I have no clue what I should do with this or withthe equation of L, somebody pointing me towards the next step would be muchappreciated J. Cheers.

 
12345678 said:
‘The line L has gradient -2 and passes through the point A (3,5). B is a point of the line L such that the distance AB is 6√5. Find thecoordinates of each of the possible points B’.
So I first calculated the equation of L…
Y - Y₁ = m (X - X₁)
Y – 5 = -2 (X – 3)
Y – 5 = -2X + 6
Y = 11 – 2X
Click to expand...

\(\displaystyle (x-3)^2+(6-2x)^2=(6\sqrt5)^2\)
 
Hello, 12345678!

I solved it "by inspection" (sort of).
The line L has gradient -2 and passes through the point A(3,5).
B is a point on line L such that the distance AB is \(\displaystyle 6\sqrt{5}\).
Find the coordinates of each of the possible points B.
Click to expand...
Code: 
        |
     B* |
       \|
        \
        |\
        | \
        |  \A
        |   *(3,5)
        |   :\
        |   : \
  - - - + - : -\- - -
        |   :   \
        |   :    \
        |   + . . *B
        |
Let point \(\displaystyle B\) be \(\displaystyle (x,y).\)

Since \(\displaystyle AB\) is \(\displaystyle 6\sqrt{5}\),
. . \(\displaystyle (x-3)^2 + (y-5)^2 \:=\:180\)

We want two squares that total 180.
I saw that \(\displaystyle 36 + 144\) works.

So we have: \(\displaystyle \begin{Bmatrix}(x-3)^2 &=& 36 \\ (y-5)^2 &=& 144 \end{Bmatrix}\)

Hence: .\(\displaystyle \begin{Bmatrix}x-3 \:=\: \pm6 & \Rightarrow & x &=& 9,\text{-}3 \\ y-5 \:=\: \pm12 & \Rightarrow & y &=& \text{-}7,\,17 \end{Bmatrix}\)

Therefore: .\(\displaystyle (9,\,\text{-}7),\;(\text{-}3,\,17)\) which are on line L.
 
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