Question

x/a + y/b + z/c = root2 and a/x + b/y + c/z = 0. find the value of x^2/a^2 + y^2/b^2 + z^2/c^2

Answer 1

$\underline{\textbf{Given:}}$

$\mathsf{\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=\sqrt{2}}$

$\mathsf{\dfrac{a}{x}+\dfrac{b}{y}+\dfrac{c}{z}=0}$

$\underline{\textbf{To find:}}$

$\textsf{The value of}$

$\mathsf{\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}}$

$\underline{\textbf{Solution:}}$

$\mathsf{Consider,}$

$\mathsf{\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}=\sqrt{2}}$

$\textsf{Squaring on bothsides, we get}$

$\mathsf{\left(\dfrac{x}{a}+\dfrac{y}{b}+\dfrac{z}{c}\right)^2=(\sqrt{2})^2}$

$\textsf{Using the identity,}$

$\boxed{\mathsf{(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca}}$

$\mathsf{\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}+2\dfrac{xy}{ab}+2\dfrac{yz}{bc}+2\dfrac{xz}{ac}=2}$

$\mathsf{\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}+2\dfrac{xyz}{abc}\left(\dfrac{a}{x}+\dfrac{b}{y}+\dfrac{c}{z}\right)=2}$

$\mathsf{\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}+2\dfrac{xyz}{abc}\left(0\right)=2}$

$\implies\boxed{\mathsf{\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}+\dfrac{z^2}{c^2}=2}}$