This is a 3-D Geometry Problem.Please help me out!P is a variable point such that its distance from the xy-plane is always equal to one-fourth of the square of its distances from the y-axis.Show that the locus of P is a cylinder.
To prove that the locus of point P is a cylinder, we need to show that all points on that locus satisfy the equation of a cylinder in 3D space.
Let's start by assuming the coordinates of point P as (x, y, z).
We are given that the distance of point P from the xy-plane is always equal to one-fourth of the square of its distance from the y-axis. Mathematically, we can express this as:
√(x^2 + y^2 + z^2) = 1/4 * (y^2)
Squaring both sides of the equation, we get:
x^2 + y^2 + z^2 = 1/16 * y^4
Bringing all terms to one side, we have:
x^2 + z^2 - 1/16 * y^4 + 1/16 * y^2 = 0
Now, let's rearrange the equation to represent a standard equation of a cylinder. We can express it as:
x^2 + z^2 = (1/16 * y^2) - (1/16 * y^4)
Let's simplify it further:
x^2 + z^2 = y^2/16 * (1 - (1/4 * y^2))
We can see that the equation resembles the standard equation of a cylinder:
x^2 + z^2 = r^2
In this case, the radius squared (r^2) is given by y^2/16 * (1 - (1/4 * y^2)). Thus, the equation represents a cylinder in 3D space.
Therefore, we have proved that the locus of point P is indeed a cylinder.