Angle Between A Line And A Plane

Given: Points A(0, 4) and B(2t, 0)

1. Midpoint of AB:
Using the midpoint formula:
Midpoint M =

2. Slope of AB:

Let be the slope of the perpendicular bisector. Then:

3. Equation of the perpendicular bisector:
Using point-slope form with point M(t, 2) and slope :

4. Intersection with y-axis (put ):


So, point R =

5. Midpoint of MR (call it point P):
M = , R =
Midpoint P =

6. Eliminate parameter t:
Let , so
Substitute in :

Final locus equation:

This is the equation of a parabola.

Answer: Option (B) : a parabola

Given the equation of the plane , the normal vector to the plane is given by the coefficients of x, y, and z:

The direction vector of the X-axis is given by:

The angle between the normal vector and the X-axis can be found using the dot product formula:

The dot product is:

The magnitudes are:

Now, we can find the cosine of the angle:

Therefore, the angle is:

Final Answer: