Question

Question :

5ˣ = 7ʸ = 1225 then find $\displaystyle \sf \left( \dfrac{xy}{x+y} \right) .$

Answer 1

Consider ,

$\sf 5^x = 1225$

$\sf \longrightarrow 5 = 1225^{\frac{1}{x}}$

Similarly ,

$\longrightarrow \sf 7 = 1225^{\frac{1}{y}}$

Now multiply these two equations ,

$\longrightarrow \sf 5 . 7 = 1225^{\frac{1}{x}}.1225^{\frac{1}{y}}$

$\longrightarrow \sf 5 . 7 = 1225^{\frac{1}{x}+\frac{1}{y}}$

$\longrightarrow \sf 35 = (35^2)^{\frac{1}{x}+\frac{1}{y}}$

$\longrightarrow \sf 35^1 = 35^{2\left(\frac{1}{x}+\frac{1}{y}\right)}$

Bases are equal , so exponents should be equal .

$\longrightarrow \sf 1 = 2\left(\dfrac{1}{x}+\dfrac{1}{y} \right)$

$\longrightarrow \sf 2\left(\dfrac{1}{x}+\dfrac{1}{y} \right)=1$

$\longrightarrow \sf \left(\dfrac{1}{x}+\dfrac{1}{y} \right)=\dfrac{1}{2}$

$\longrightarrow \sf \left(\dfrac{x+y}{xy} \right)=\dfrac{1}{2}$

$\longrightarrow \sf \blue{\left(\dfrac{xy}{x+y} \right)=2}$