Question

Let $A\equiv (a,b)$ and $B\equiv (c,d)$ where $c>a>0$ and $d>b>0$. Then point $C$ on the $x-$axis such that $AC+BC$ is minimum,is

• A. $\frac{bc-ad}{b-d}$
• B. $\frac{ac+bd}{b+d}$
• C. $\frac{ac-bd}{b-a}$
• D. $\frac{ad+bc}{b+d}$

Answer 1

The correct option is D $\frac{ad+bc}{b+d}$

$A^{\prime }$ is the image of $A$ in the $x-$axis


So, the coordinates of $A^{\prime }\equiv (a,-b)$
for minimum value of $AC+BC,A^{\prime }C+BC$ should be minimum $\because AC=A^{\prime }C$
For $A^{\prime }C+BC$ to be minimum , $A^{\prime },B,C$ should be colinear
$\therefore$ equation $A^{\prime }B$ will be
$y+b=\frac{d+b}{c-a}(x-a)$
So, for coordinate of $C$ put $y=0$ in the above equation
$bc-ba=x(b+d)-ad-ab$
$\Rightarrow x=\frac{cb+ad}{b+d}$