Question

(x-a) (x-b) /(c-a) (c-b) +(x-b) (x-c) /(a-b) (a-c)+(x-c) (x-a) /(b-c) (b-a) =1​

Answer 1

Answer:

(x-a)(x-b)+(x-b)(x-c)+(x-c)(x-a)=0

x^2-(a+b)x+ab+x^2-(b+c)x+bc+x^2-(c+a)x+ca=0

3x^2–2(a+b+c)x+(ab+bc+ca)=0

Roots are equal , B^2–4AC=0

[-2(a+b+c)]^2 -4×3×(ab+bc+ca)=0

4(a+b+c)^2=4×3×(ab+bc+ca)

(a+b+c)^2=3(ab+bc+ca)

a^2+b^2+c^2+2(ab+bc+ca)=3(ab+bc+ca)

a^2+b^2+c2=ab+bc+ca……………..(1)

Now put a=b=c = (k) let

k^2+k^2+k^2=kk+kk+kk

3k^2=3k^2 , (true) . Proved.