Question

A variable line having negative slope and passing through the point $(8,2)$ meets the axes at $A$ and $B$. Find the minimum value of the sum $OA+OB$ where $O$ is the origin.

• A. $9$
• B. $18$
• C. $27$
• D. $36$

Answer 1

The correct option is B

$18$

Equation of any line through $(8,2)$ can be chosen as
$y-2=m(x-8)$
which meets the axes at points
$A(8-\frac{2}{m},0)$ and $B(0,2-8m)$
Thus, we have
$OA+OB=10-8m-\frac{2}{m}=f\left(m\right)$ (say)
the value of $m$ at which $f$ attains minima is given by
$\frac{df}{dm}=-8+\frac{2}{m^2}=0,$ i.e.,$m=\pm \frac{1}{2}$
Now,$\frac{d^{2}f}{d{m}^2}=\frac{-4}{m^3}$ is positive for $m=-\frac{1}{2}$. hence, the value of $f$ is minimum at $m=\frac{-1}{2}$.
Hence, the minimum value of $OA+OB$ is $18$