Find Co-ordinates of Point C in Problem Involving Straight Line Equations

AI Thread Summary
The discussion focuses on finding the coordinates of point C based on straight line equations involving points A and B. The initial approach uses gradients and the equation of a circle to derive the coordinates, resulting in C being either (3, 0) or (-1, 8). A simpler method is proposed, emphasizing that the segment from B to C is perpendicular to AB and has the same length, leading to the same coordinates for C. The conversation acknowledges the effectiveness of both methods while noting some initial ambiguity in the simpler approach. Ultimately, both methods confirm the coordinates of point C as (3, 0) and (-1, 8).
chwala
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Homework Statement
See attached
Relevant Equations
straight line equations
Find the question here; My interest is on question ##3(c)## only.

1647999147492.png


My approach, Let the co ordinates of ##C##= ##(x,y)## then considering points ##B## and ##C##. We shall have the gradient given by;

##\dfrac {y-4}{x-1}##=##-2##

also from straight line equation, considering points ##A## and ##C##, we shall have;
##(x+3)^2+(y-2)^2=40##
we know that, ##y=-2x+6## from the given equations above, then we shall have,
##(x+3)^2+(-2x+6-2)^2=40##
##(x+3)^2+(-2x+4)^2=40##
##5x^2-10x-15=0##
##x^2-2x-3=0##
therefore possible co ordinates of ##C## are ##(3,0)## and ##(-1,8)##

I am seeking a much simpler approach...of course i assume the reader is conversant with my approach...because of time i cannot show step by step...but shout out to me if an equation is not clear. Bingo! :cool:heeey!
 
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Simpler approach:
To get from A to B we add 4 to x and 2 to y.
The segment from B to C is perpendicular to AB and the same length, so it must have absolute values of changes in x and y reversed, ie 2 and 4, and those changes must have opposite signs.
Hence, adding those changes to the coordinates of B = (1,4), we see it will be
(1 + 2, 4 - 4) = (3, 0)
OR
(1 - 2, 4 + 4) = (-1, 8)
 
andrewkirk said:
Simpler approach:
To get from A to B we add 4 to x and 2 to y.
The segment from B to C is perpendicular to AB and the same length, so it must have absolute values of changes in x and y reversed, ie 2 and 4, and those changes must have opposite signs.
Hence, adding those changes to the coordinates of B = (1,4), we see it will be
(1 + 2, 4 - 4) = (3, 0)
OR
(1 - 2, 4 + 4) = (-1, 8)
Thanks, I had initially thought of this approach and found it ambiguous ...correct though...
 
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