Geometry

We are given the line and the point .
We need to find the coordinates of the foot of the perpendicular from point to this line.
Step 1: Equation of the perpendicular line
The slope of the given line is , so the slope of the perpendicular line will be , since the product of the slopes of two perpendicular lines is . The equation of the line passing through with slope is: Simplifying:
Step 2: Solve the system of equations
We now have the system of equations:
1.

2.

Substitute the second equation into the first:
Step 3: Solve for and
After simplifying and solving for and , we find that the foot of the perpendicular is at . Thus, the correct answer is , corresponding to option (a).

We are given two points and , and a point .
We need to find the inclination of the straight line passing through and the midpoint of the line joining and .
Step 1: Find the midpoint of and
The midpoint of the line joining and is given by: Substituting the coordinates of and , we get:
Step 2: Find the slope of the line through and
The slope of the line through points and is given by: Substituting the coordinates of and , we get:
Step 3: Find the inclination
The inclination of a line with slope is given by: Substituting , we get: Since the inclination is typically expressed as an acute angle, we take the positive value of the angle: Thus, the correct answer is , corresponding to option (b).