Question

Assertion :If $x^2+9y^2+25z^2=xyz(\frac{15}{x}+\frac{5}{y}+\frac{3}{z})$, then x, y, z are in H.P. Reason: If $a_1^2+a_2^2+\cdots +a_n^2=0$, then $a_1=a_2=a_3=\cdots =a_n=0$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is A Both Assertion and Reason are correct and Reason is the correct explanation for Assertion

$x^2+9y^2+25z^2=xyz(\frac{15}{x}+\frac{5}{y}+\frac{3}{z})$

$\Rightarrow x^2+9y^2+25z^2-15yz-5xz-3xy=0$

$\Rightarrow 2x^2+18y^2+50z^2-30yz-10xz-6xy=0$

$\Rightarrow (x-3y{)}^2+(3y-5z{)}^2+(5z-x{)}^2=0$

$\Rightarrow x-3y=0,3y-5z=0,5z-x=0$

$\Rightarrow x=3y=5z=k$ (say)

$\Rightarrow x=k,y=\frac{k}{3},z=\frac{k}{5}$

Hence, $x,y,z$ are in H.P. Hence option (a) is correct.