Question

Let a, b and c be non-zero real numbers satisfying (a ^ 3)/(b ^ 3 + c ^ 3) + (b ^ 3)/(c ^ 3 + a ^ 3) + (c ^ 3)/(a ^ 3 + b ^ 3) = 1; then the value of
a^ 6 b^ 3 +c^ 3 + b^ 6 c^ 3 +a^ 3 + c^ 6 a^ 3 +b^ 3 is
a.1
b. 0
c. -1
d.2​

Answer 1

Given :- Let a, b and c be non-zero real numbers satisfying (a³)/(b³ + c³) + (b³)/(c³ + a³) + (c³)/(a³ + b³) = 1 .

Then the value of :- (a⁶)/(b³ + c³) + (b⁶)/(c³ + a³) + (c⁶)/(a³ + b³) = ?

A) 1

B) 0

C) -1

D) 2

Solution :-

we have,

→ (a³)/(b³ + c³) + (b³)/(c³ + a³) + (c³)/(a³ + b³) = 1

Let :-

• a³ = x
• b³ = y
• c³ = z

so,

→ x/(y + z) + y/(x + z) + z/(x + y) = 1 -------- Eqn.(1)

and,

→ (a⁶)/(b³ + c³) + (b⁶)/(c³ + a³) + (c⁶)/(a³ + b³)

→ x²/(y + z) + y²/(x + z) + z²/(x + y)

adding and subtracting x,y and z in each terms,

→ [x²/(y + z) + x - x] + [y²/(x + z) + y - y] + [z²/(x + y) + z - z]

→ [(x² + xy + yz)/(y + z) - x] + [(y² + xy + yz)/(x + z) - y] + [(z² + xz + yz)/(x + y) - x]

→ {x(x + y + z)/(y + z)} + {y(x + y + z)/(x + z)} + {z(x + y + z)/(x + y)} - (x + y + z)

taking (x + y + z) common

→ (x + y + z)[x/(y + z) + y/(x + z) + z/(x + y) - 1]

putting value from Eqn.(1) ,

→ (x + y + z)(1 - 1)

→ 0 (B) (Ans.)

Learn more :-

if a^2+ab+b^2=25

b^2+bc+c^2=49

c^2+ca+a^2=64

Then, find the value of

(a+b+c)² - 100 = ...

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JEE mains Question :-

https://brainly.in/question/22246812