The plane x+2y+z=6 cuts x,y and z axes at A,B and C.Find x coordinate of circumcentre of ∆ABC

Question

priyar · Answer

you can  find the three coordinates.. then use that to find the circumcentre

samigupta8 · Answer

Really! do you know the formula for that?
For circumcentre part

priyar · Answer

no i don't know any formula but there are two methods..

samigupta8 · Answer

One method i definitely tried.Bt ended up nowhere.

priyar · Answer

which one ? intersection of perpendicular bisectors?

samigupta8 · Answer

The way in which we equate the radius of the circle knowing the three vertices and a centre assumed to be (a,b,c)

priyar · Answer

oh okay, but the shorter method is you can find angle A and the solve

samigupta8 · Answer

Other way is finding or tho center but i don't know that too.

priyar · Answer

there is one formula that involves sin2A.. have you heard of it?

priyar · Answer

but the perpendicular bisectors intersection will surely work

samigupta8 · Answer

We can't apply it here ,i think so (sin2A formula) becoz this is a 3-D figure.

samigupta8 · Answer

I m just now getting how can you apply perpendicular bisectors also becoz we never know the equation of lines here.
Do we?

samigupta8 · Answer

I m just now getting how can you apply perpendicular bisectors also becoz we never know the equation of lines here.
Do we?

parthkohli · Answer

$$(6,0,0), (0, 3, 0), (0, 0, 6)$$By symmetry, the $x$ and $z$ coordinates of the center will be the same. So assume the center to be $(a, b, a)$.$$(a-6)^2 + b^2 + a^2 = a^2 + (b-6)^2 + a^2 $$$$\Rightarrow -6(2a-6) = 6(2b-6)$$$$\Rightarrow 3 - a = b-3$$$$\Rightarrow a + b = 6$$$$(a-6)^2 + (a-6)^2+a^2 = a^2+(a-3)^2 + a^2 $$Finally, $a = 7/2$ and $b = 5/2$.

The circumcenter is $(7/2, 5/2, 7/2)$

priyar · Answer

we can find the eqs of line using the coordinates

priyar · Answer

If you want to know this method (per. bis.) here is a link: https://www.youtube.com/watch?v=kwooTzS01PA

samigupta8 · Answer

How you did this?
a^2+(b-6)^2+a^2 in the first equation
I got this a^2+(b-3)^2+a^2

samigupta8 · Answer

@parthkohli i literally don't know how did you came up with values of a and b 
I just got an equation which is 9=4a-2b

priyar · Answer

Even i didn't understand.. @parthkohli can explain from the symmetry part?

samigupta8 · Answer

Did you get the same equation as i wrote above.
Further as far as i see,he has done a typo in that.

priyar · Answer

Even i didn't understand.. @parthkohli can explain from the symmetry part?