Question

If you know the answer then only write ​

Answer 1

Given:-

• if $\rm{\dfrac{x}{y} + \dfrac{y}{x} = -1}$

To Find:-

• The value of x³ - y³

Identities used:-

• x³ - y³ = ( x - y ) ( x² + xy + y² )

Now,

→ $\rm{\dfrac{x}{y} + \dfrac{y}{x} = -1}$

→ The Lcm of x and y is xy

Therefore,

→ $\rm{\dfrac{x}{y} + \dfrac{y}{x} = -1}$

→ $\rm{\dfrac{x² + y²}{xy} = -1 }$

→ $\rm{ x² + y² = -xy }$

Now,

→ x³ - y³ = ( x - y) ( x² + y² + xy )

→ it is being proved that x² + y² = -xy

→ x³ - y³ = ( x - y ) ( -xy + xy )

→ x³ - y³ = ( x - y ) ( 0 )

→ x³ - y³ = 0

Hence, The Value of x³ - y³ is 0

Hence, C is correct option.

Now, Coming to Second Question

Given:-

• if a + b + c = 0

To Find:-

• $\rm{\dfrac{a^2}{bc} + \dfrac{b^2}{ca} + \dfrac{c^2}{ab}}$

Concept used:-

• if a + b + c = 0 then a³ + b³ + c³ = 3abc

Now,

→ $\rm{\dfrac{a^2}{bc} + \dfrac{b^2}{ca} + \dfrac{c^2}{ab}}$

→ The LCM of bc, ca and ab is abc

→ $\rm{\dfrac{a^2}{bc} + \dfrac{b^2}{ca} + \dfrac{c^2}{ab}}$

→ $\rm{\dfrac{a^3 + b^3 + c^3}{abc}}$

→ $\rm{\dfrac{3abc}{abc}}$

→ $\rm{3}$

Therefore, The value is 3.

Hence, D is correct.