Coordinate Geometry

For three points to be collinear, the slope between the first and second points must be equal to the slope between the second and third points. Using the slope formula: Setting the two slopes equal: Solving this gives . Thus, the correct answer is 1.

The distance between two points and is given by: Substituting the given points: Using the identity for the difference of cosines and sines: Thus, the correct answer is .

The formula to calculate the distance between two parallel lines and is: Substituting the values for the given lines and : Thus, the correct answer is unit.

To find the coordinates of point P dividing the line segment AB in the ratio 1:2, we use the section formula:

1. The section formula for a point dividing a segment joining and in ratio is:
2. Given: A(2, 3), B(8, 9), ratio 1:2 ( , ).
3. Calculate the x-coordinate:
4. Calculate the y-coordinate:
5. The coordinates of P are (4, 5).
6. Match with options: (4, 5) corresponds to option (A).

Thus, the correct answer is: