For the three points $A(2,0), B(0,2)$ and $P(1,1)$, suppose $d$ is the algebraic sum of the distances of $A$…
For the three points $A(2,0), B(0,2)$ and $P(1,1)$, suppose $d$ is the algebraic sum of the distances of $A$ and $B$ from a line that passes through $P$. Then, which of the following is correct?
$d>0$ for all lines
$d=0$ for at least one line
$d=0$ for all lines
$d>0$ at least for one line
Solution
Let the variable line be $a x+b y+c=0$ The line passes through $(1,1)$, then
$
a+b+c=0...(i)
$
$d=$ algebraic sum of distance from $A$ and $B$ to the line
$
\begin{aligned}
d & =\left|\frac{2 a+0 b+c}{\sqrt{a^2+b^2}}\right|+\left|\frac{0 a+2 b+c}{\sqrt{a^2+b^2}}\right| \\
& =\frac{2(a+b+c)}{2 \sqrt{a^2+b^2}} \\
& =0 [ \because From Eq. (i)]
\end{aligned}
$
$\therefore \quad d=0, \forall$ all the lines