If $(a - b)^{2} + (b - c)^{2} + (c - a)^{2} = 0$ , a,b,c ∈ R and a:b:c = 1:m:n, find m+n

__

Open in App

Solution

If $x^{2} + y^{2} + z^{2} = 0$ , x,y,z $ϵ$ R

Then x=0,y=0,z=0

$(a - b)^{2} + (b - c)^{2} + (c - a)^{2} = 0$

$\Rightarrow$ a-b = 0, b-c =0, c-a =0

$\Rightarrow$ a=b=c

$\Rightarrow$ a:b:c = 1:1:1

$\Rightarrow$ m=1,n=1

m+n =2

Suggest Corrections

Similar questions

a = 1

b = - 1

c = 0

(a^{2} - b^{2}) + (c^{2} - b^{2}) + (a^{2} - c^{2})

a + b + c = 0

\frac{1}{b^{2} + c^{2} - a^{2}} + \frac{1}{c^{2} + a^{2} - b^{2}} + \frac{1}{a^{2} + b^{2} - c^{2}}

\in R^{+}

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = 1

\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1

- \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} + \frac{z^{2}}{c^{2}} = 1

\frac{a^{2} - b^{2} - c^{2}}{(a - b) (a - c)} + \frac{b^{2} - c^{2} - a^{2}}{(b - c) (b - a)} + \frac{c^{2} - a^{2} - b^{2}}{(c - a) (c - b)}

a = 1

b = - 1

c = 0

a^{2} + b^{2} + c^{2}

Join BYJU'S Learning Program