Question

 what is the value of  \[\frac{a^2}{bc} + \frac{b^2}{ca} + \frac{c^2}{ab}\] .

 

Answer 1

The co-ordinates of the vertices are (a, b); (b, c) and (c, a)

The co-ordinate of the centroid is (0, 0)

We know that the co-ordinates of the centroid of a triangle whose vertices are `(x_1 , y_1 ) ,(x_2 ,y_2 ) , (x_3 , y_3)` is-

`((x_1 + x_2 + x_3 ) / 3  , (y_1 + y_2 +y_3 ) / 3)`

So,

`( 0 , 0) = ((a + b +c ) / 3 , ( b +c +a ) / 3 )`

Compare individual terms on both the sides-

`(a + b+ c ) / 3 = 0`

Therefore,

a + b + c = 0

We have to find the value of -

=`(a^2)/(bc) + (b^2)/(ca) + (c^2)/(ab) ` 

Multiply and divide it by (abc)  to get,

` = (1/(abc)) ( a^3 + b^3 + c^3 )`

Now as we know that if,

a  + b + c = 0

Then,

`a^3 + b^3 + c^3 = 3abc`

So,

`(a^2 ) /( bc) +(b^2)/(ca) + (c^2)/(ab) = (1/(abc)) (a^3 + b^3 +c^3)`

                            `= (1/(abc))(3abc)`

                             = 3