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Hey guys new to the forum hoping you can fix my stupidity. SO the first problem asks to find the equation of the circle whose center is on the y axis and that contain the points (1,4) and (-3,2). Give the center and the radius. I can find the radius and write the equation I am just struggling to find the midpoint. The other problem is similar. Find the midpoint of a circle passing through the points (3,1) (0,0) and (8,4). Any help would be greatly appreciated!
So if your center is on the y axis it means the x coordinate is 0 and since the distance from center of circle to the points it contains is equal, if (0, y) was our center.
Distance between (0,y) and (1,4) = Distance between (0,y) and (-3,2)
So using the distance formula. Distance between (0,y) and (1,4) = $\displaystyle \sqrt{1 + (y-4)^2} $
Distane between (0,y) and (-3,2) is $\displaystyle \sqrt{9 + (y-2)^2} $
so $\displaystyle \sqrt{1 + (y-4)^2} = \sqrt{9 + (y-2)^2} $ or $\displaystyle 1 + (y-4)^2 = + (y-2)^2 $
which gives the identity, $\displaystyle -4 + 4y = 0 $ or y = 1
so (0,1) is your center and $\displaystyle \sqrt{9 + (1-2)^2} = \sqrt(10) $ is your radius.
For the second one.
Do the same thing so you should get 2 linear equations in 2 unknown. If (x,y) were our midpoint
Distance between (x,y) and (3,1) = Distance between (x,y) and (0,0)
and
Distance between (x,y) and (8,4) = Distance between (x,y) and (0,0) (Note: You could also use (3,1) here but (0,0) is easier)
So Distance b/w (x,y) and (3,1) = $\displaystyle \sqrt{(x-3)^2 + (y-1)^2}$
Distance b/w (x,y) and (0,0) = $\displaystyle \sqrt(x^2 + y^2) $
so $\displaystyle \sqrt{(x-3)^2 + (y-1)^2} = \sqrt(x^2 + y^2) $ or $\displaystyle x-3)^2 + (y-1)^2 = x^2 + y^2 $ or $\displaystyle y = 5-3x $
Now doing the same for Distance b/w (x,y) and (8,4) = $\displaystyle \sqrt{(x-8)^2 + (y-4)^2}$
Distance b/w (x,y) and (0,0) = $\displaystyle \sqrt{x^2 + y^2} $
setting them equal $\displaystyle (x-8)^2 + (y-4)^2 = x^2 + y^2 $ or $\displaystyle y = 10 - 2x $
solve $\displaystyle y = 10 - 2x $ and $\displaystyle y = 5 - 3x $ or 10-2x = 5 - 3x and so x = -5 and y = 20
our circle center (-5,20)
I was wondering if you could check me on this next problem. Same layout. Find the circle with the points (1,-4)(-3,4)(4,5) I came up with Center (1,1) radius of 5. I would show you all my work but it I am new to the site and it would take me forever. It seems to check out because the points are all 5 units from center. Just wanted to double check with someone. Thanks in advance.
Looks good
