To solve the given problem, we need to find the value of the expression:
\(\frac{x^2}{yz} + \frac{y^2}{zx} + \frac{z^2}{xy}\)
given that \(x + y + z = 0\).
We can rewrite this condition as:
\(z = -(x + y)\)
Substituting \(z = -(x + y)\) into the expression gives us:
\(\frac{x^2}{y(-x-y)} + \frac{y^2}{x(-x-y)} + \frac{(-x-y)^2}{xy}\)
Simplifying each fraction:
\(\frac{x^2}{y(-x-y)} = -\frac{x^2}{y(x+y)}\)
\(\frac{y^2}{x(-x-y)} = -\frac{y^2}{x(x+y)}\)
\(\frac{(-x-y)^2}{xy} = \frac{x^2 + 2xy + y^2}{xy}\)
Thus, the expression can be rewritten as:
\(-\frac{x^2}{y(x+y)} - \frac{y^2}{x(x+y)} + \frac{x^2 + 2xy + y^2}{xy}\)
We can combine the first two terms:
\(-\left(\frac{x^3}{xy(x+y)} + \frac{y^3}{xy(x+y)}\right) = -\frac{x^3 + y^3}{xy(x+y)}\)
Using the identity \(x^3 + y^3 = (x+y)(x^2 - xy + y^2)\), and since \(z = -(x+y)\), we get:
\(x^3 + y^3 = -(x^2 - xy + y^2)(x+y)\)
So, the expression becomes:
\(\frac{x^2 + 2xy + y^2 - (x^2 - xy + y^2)}{xy} = 3\)
Hence, the final value of the expression is 3.
The correct option is 3.
Similar Questions
- If $a + b + c = 10$ and $ab + bc + ca = 31$, find the value of $a^2 + b^2 + c^2$
If $a + b + c = 0$, then find the value of $\frac{(a^2+b^2+c^2)^2}{a^2b^2+b^2c^2+c^2a^2}$
- If $x^4 + \frac{1}{x^4} = 322$, then $x^3 - \frac{1}{x^3} = ?$
- If $(x + \frac{1}{x}) = 7$, then $(x - \frac{1}{x})$ is equal to:
- What is the value of the following expression:
$(1 + x)(1 + x^2)(1 + x^4)(1 + x^8)(1 - x)$ - If $a + b + c = 17$, $abc = 168$, and $ab + bc + ca = 94$, then $a^3 + b^3 + c^3 = ?$
- If $a = \frac{b^2}{b - a}$, then the value of $a^3 + b^3$ is:
- If $ab = 10$ and $a^{2} + b^{2} = 29$, then $(a - b)^{2} = ?$
- Find the value of $\frac{(74 + 47)^2 + (74 - 47)^2}{74^2 + 47^2}$
- If $\left(\text{a}^2 + \frac{1}{\text{a}^2}\right) = 18$ then find $\left(\text{a} - \frac{1}{\text{a}}\right)$
Important Questions from Identities
If x 2 + 4y 2 = 40, xy = 6 and x > 2y then the value of x - 2y is:
- \(\dfrac{(5.17-2.19)^2-(5.17+2.19)^2}{11.3223}\)
If a(a + b + c) 2= 1792; b(a + b + c) 2= 1536; c(a + b + c) 2= 768 then what will be the value of a?
If a + b + c = 5, a 2+ b 2+ c 2= 27, a 3 + b 3 + c 3 = 125, then the value of \(\frac{abc}{5}\) is:
(999) 2- 2 2=
