Question

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?

• A. \(d\gt 0\) for all lines
• B. \(d=0\) for at least one line
• C. \(d=0\) for all lines
• D. \(d\gt 0\) at least for one line

Hint:

If a line passes through the midpoint of two points, then the algebraic distances of those two points from the line are equal in magnitude and opposite in sign.

Answer 1

The Correct Option is C

Step 1: Observe the given points.
The given points are \[ A(2,0),\quad B(0,2),\quad P(1,1) \]

Step 2: Check the midpoint of \(A\) and \(B\).
The midpoint of \(A(2,0)\) and \(B(0,2)\) is \[ \left(\frac{2+0}{2},\frac{0+2}{2}\right) \] \[ =(1,1) \] Therefore, \[ P(1,1) \] is the midpoint of \(AB\).

Step 3: Consider any line passing through \(P\).
Let any line passing through \(P(1,1)\) be \[ L=0 \] Since \(P\) is the midpoint of \(AB\), the points \(A\) and \(B\) are symmetric with respect to \(P\).

Step 4: Use algebraic distance concept.
For any line passing through the midpoint \(P\), the algebraic distances of \(A\) and \(B\) from that line are equal in magnitude and opposite in sign.
So, if the algebraic distance of \(A\) from the line is \[ r, \] then the algebraic distance of \(B\) from the same line is \[ -r \]

Step 5: Find the algebraic sum.
The algebraic sum of distances is \[ d=r+(-r) \] \[ d=0 \]

Step 6: Check whether this is true for every line.
The above argument depends only on the fact that the line passes through the midpoint \(P\).
Therefore, for every line passing through \(P\), the algebraic distances of \(A\) and \(B\) cancel each other.

Step 7: Final conclusion.
Hence, \[ d=0 \] for all lines passing through \(P\).
Therefore, \[ \boxed{d=0\text{ for all lines}} \]