Question

if a,b,c is not equal to 0 and a+b+c=0 prove that a²/bc+b²/ca+c²/ab=3​

Answer 1

Answer :-

Step-by-step explanation :-

Given :-

→ a + b + c = 0 .

We know that ,

→ If a + b + c = 0, then a³ + b³ + c³ = 3abc .

[ Because , a³ + b³ + c³ - 3abc = ( a + b + c ) ( a² + b² + c² - ab - bc - ca ) .

→ a³ + b³ + c³ - 3abc = ( 0 ) ( a² + b² + c² - ab - bc - ca ) .

→ a³ + b³ + c³ - 3abc = 0 .

→ a³ + b³ + c³ = 3abc . ]

Now, To prove :-

$\sf \because \frac{ {a}^{2} }{bc} + \frac{ {b}^{2} }{ca} + \frac{ {c}^{2} }{ab} = 3.$

Solution :-

$\sf \because \frac{ {a}^{2} }{bc} + \frac{ {b}^{2} }{ca} + \frac{ {c}^{2} }{ab} . \\ \\ \sf = \frac{ {a}^{3} + {b}^{3} + {c}^{3} }{abc} . \\ \\ \sf = \frac{3 \cancel{abc}}{ \cancel{abc} }. \\ \\ \large \it = 3. \\ \\ \large \pink{ \boxed{ \boxed{ \mathscr{ \therefore \underline{ LHS = RHS }}}}}$

Hence, it is proved .