Distance Formula

Step 1: Recall the section formula.
The coordinates of a point that divides the line segment joining and internally in the ratio are given by the section formula: However, the standard internal division formula with
(where is the part towards and towards ) is: where and are the ratios from to and to . Step 2: Interpret the ratio .
The ratio means divides such that the segment from to is parts and from to is parts. The correct formula for internal division is: But the standard form with as the ratio in which divides internally from to is: which seems reversed. The correct interpretation is: where is the part of , is the part of . Step 3: Match with options.
Option (1): ,
This matches the standard section formula where is the ratio in which divides internally, with associated with and with .
Options (2), (3), and (4) have different combinations or signs, which do not align with the internal division formula.
Step 4: Verify the formula.
For example, if , (ratio 1:1, midpoint): which is the midpoint, confirming option (1). Step 5: Select the correct answer.
The coordinates are , matching option (1).

Step 1: Use the distance formula.
The distance between two points and is given by: Here, the points are and , so , , , . Step 2: Compute the differences.
,
.
Step 3: Calculate the distance.
Simplify : Step 4: Correct the calculation.
Recompute carefully: However, let’s verify the coordinates and options:
,
,
,
but option (3) is . Recheck distance:
,
Possible typo in options or points. If intended and :
,
which matches option (3).
Step 5: Select the correct answer.
Assuming a possible typo in the second point (e.g., instead of ), the distance fits option (3). With given , it’s , but answer (3) suggests correction to points.