A (-6,2) and B (-4,-2) are endpoints of a chord of a circle. C (2.-4) and D (8,4) are endpoints of a second chord.
a) Determine the coordinates of the center of the circle.
b) Determine the radius of the circle.
So lost, please help
A (-6,2) and B (-4,-2) are endpoints of a chord of a circle. C (2.-4) and D (8,4) are endpoints of a second chord.
a) Determine the coordinates of the center of the circle.
b) Determine the radius of the circle.
So lost, please help
An alternative way is...
The centre of the circle is the same distance away from all 4 points.
If the centre is (x,y) then we can use the distance formula
on any 3 of the 4 points to find (x,y).
$\displaystyle [x-(-6)]^2+[y-2]^2=[x-(-4)]^2+[y-(-2)]^2=[x-2]^2+[y-(-4)]^2$
$\displaystyle (x+6)^2+(y-2)^2=(x+4)^2+(y+2)^2=(x-2)^2+(y+4)^2$
Multiply this out
$\displaystyle x^2+12x+36+y^2-4y+4=x^2+8x+16+y^2+4y+4=x^2-4x+4+y^2+8y+16$
Since $\displaystyle x^2+y^2$ is common to all three, we can eliminate them
$\displaystyle 12x-4y+40=8x+4y+20=-4x+8y+20$
Combining the first two equations to write x in terms of y
$\displaystyle 12x-8x+40=4y+4y+20\ \Rightarrow\ 4x+40=8y+20\ \Rightarrow\ x+10=2y+5\ \Rightarrow\ x=2y-5$
Using this in the 2nd and 3rd equations
$\displaystyle 8(2y-5)+4y+20=-4(2y-5)+8y+20$
$\displaystyle 16y-40+4y+20=-8y+20+8y+20$
$\displaystyle 20y-20=40$
$\displaystyle 20y=60\ \Rightarrow\ y=3\ \Rightarrow\ x=6-5=1$
The radius is the distance from the centre to any of the 4 points
$\displaystyle r=\sqrt{(8-1)^2+(4-3)^2}=\sqrt{7^2+1^2}=\sqrt{50}$
Thank you so, so much! I wish I could hug you!
I'm a Mom of three, taking a grade 12 math class at night. I haven't been in a classroom in over 15 years and I'm finding it very challenging (to put it nicely lol).
This forum has been invaluable to me.
Distance from (1,3) to (-6,2) is $\displaystyle \sqrt{7^2+1^2}$
Distance from (1,3) to (-4,-2) is $\displaystyle \sqrt{5^2+5^2}$
Distance from (1,3) to (2,-4) is $\displaystyle \sqrt{1^2+7^2}$
Distance from (1,3) to (8,4) is $\displaystyle \sqrt{7^2+1^1}$
In all cases the distance is $\displaystyle \sqrt{50}$
hence both are chords of the same circle centred at (1,3) with radius $\displaystyle \sqrt{50}$
[quote=Archie Meade;469199]Distance from (1,3) to (-6,2) is $\displaystyle \sqrt{7^2+1^2}$
Distance from (1,3) to (-4,-2) is $\displaystyle \sqrt{5^2+5^2}$
Distance from (1,3) to (2,-4) is $\displaystyle \sqrt{1^2+7^2}$
Distance from (1,3) to (8,4) is $\displaystyle \sqrt{7^2+1^1}$
In all cases the distance is $\displaystyle \sqrt{50}$
hence both are chords of the same circle centred at (1,3) with radius $\displaystyle \sqrt{50}$[/quote
Sorry I'm wrong. Iplotted the points and got slightly different radii lenghts and although th center was 1,3 i should have calculated all the radii