Math, asked by venkataharshitha, 3 months ago

6. If log(x+y÷3)=1÷2(logx+logy),then find the value of x÷y+y÷x?

Answers

Answered by ItzAshi

157

Step-by-step explanation:

${\large{\mathbf{\underline{\blue{Question :→}}}}} \\$

If log (x + y)/3 = 1/2(log x + log y) then find the value of x/y + y/x

${\large{\mathbf{\underline{\blue{Solution :→}}}}} \\$

We have,

${\bold{\sf{log \: \frac{x \: + \: y}{3} \: = \: \frac{1}{2}(log x \: + \: log y)}}} \\ \\$

${\bold{\sf{: \: ⟹ \: \: \: \: \: log \frac{x \: + \: y}{3} \: \: = \: \: (log x) ^ \frac{1}{2} \: + \: (log y) ^ \frac{1}{2}}}} \\ \\$

${\bold{\sf{: \: ⟹ \: \: \: \: \: log \frac{x \: + \: y}{3} \: \: = \: \: log √x \: + \: log √y}}} \\ \\$

${\bold{\sf{: \: ⟹ \: \: \: \: \: log \frac{x \: + \: y}{3} \: \: = \: \: log \sqrt{x} \: \sqrt{y} }}} \\ \\$

${\bold{\sf{: \: ⟹ \: \: \: \: \: \frac{x \: + \: y}{3} \: \: = \: \: \sqrt{xy}}}} \\ \\$

${\bold{\sf{: \: ⟹ \: \: \: \: \: x \: + \: y \: \: = \: \: 3 \: \sqrt{xy}}}} \\ \\$

On squaring both the sides we will get,

${\bold{\sf{: \: ⟹ \: \: \: \: \: (x \: + \: y)² \: \: = \: \: 9xy}}} \\ \\$

${\bold{\sf{: \: ⟹ \: \: \: \: \: x² \: + \: y² \: + \: 2xy \: \: = \: \: 9xy}}} \\ \\$

${\bold{\sf{: \: ⟹ \: \: \: \: \: x² \: + \: y² \: = \: 9xy \: - \: 2xy}}} \\ \\$

${\bold{\sf{\red{: \: ⟹ \: \: \: \: \: }}}}{\bold{\mathfrak{\fbox{\orange{ x² \: + \: y² \: = \: 7xy}}}}} \\ \\$

On dividing both sides by xy we will get,

${\bold{\sf{: \: ⟹ \: \: \: \: \: \frac{x² \: + \: y²}{xy} \: \: = \: \: \frac{7xy}{xy}}}} \\ \\$

${\bold{\sf{\red{: \: ⟹ \: \: \: \: \: }}}}{\bold{\mathfrak{\boxed{\purple{ \frac{x}{y} \: + \: \frac{y}{x} \: \: = \: \: 7}}}}} \\$

Answered by muskansingh3707126