Question

If the sum of two numbers $p$ and $q$ is $\sqrt{10}$ and their difference is $\sqrt{6}$, then the value of ${\log }_{q}p$ is

• A. $-1$
• B. $0$
• C. $1$
• D. $2$

Answer 1

The correct option is A $-1$

Given conditions,
$p+q=\sqrt{10}\Rightarrow (p+q{)}^2=10\Rightarrow p^2+q^2+2pq=10\cdots (1)$
$p-q=\sqrt{6}\Rightarrow p^2+q^2-2pq=6\cdots \left(2\right)$
Subtracting equation $\left(2\right)$ from $\left(1\right)$, we get
$4pq=4$
$\Rightarrow pq=1$
$\Rightarrow p=\frac{1}{q}$
Taking ${\log }_q$ on both sides, we get
${\log }_{q}p={\log }_q\frac{1}{q}=-1$