Question

Two points $A\left(x_1,y_1\right)$ and $B\left(x_2,y_2\right)$ are chosen on the graph of $f(x)=\ell nx$ with $0<x_1<x_2$. The points $C$ and $D$ trisect line segment $AB$ with $AC<CB$. Through $C$ a horizontal line is drawn to cut the curve at $E\left(x_3,y_3\right)$. If $x_1=1$ and $x_2=1000$ then the value of $x_3$ equals

• A. 10
• B. $\sqrt{10}$
• C. $(10{)}^{2/3}$
• D. $(10{)}^{1/3}$

Answer 1

Answer: (A)

Using section formula
$a=\frac{2\cdot 1+1\cdot 1000}{3}=334$
$b=\frac{2\cdot 0+1\cdot \ell n1000}{3}$
$\Rightarrow b=\frac{\ell n1000}{3}...(i)$
Now line $y=b$ intersects the curve $y=\ln x$
$\therefore b=\ln x...(ii)$

from (i) and (ii)
$\frac{\ell n1000}{3}=\ln x$
$\Rightarrow \ell n(1000{)}^{1/3}=\ell nx$
$\therefore x=(1000{)}^{1/3}=10$