91Ó°ÊÓ

Understanding the distance formula is crucial for solving problems involving distances between points in three-dimensional space. The distance between two points, say \( P(x_1, y_1, z_1) \)and \( Q(x_2, y_2, z_2) \), is calculated using the distance formula:

• Step 1: Find the differences in the x, y, and z coordinates, i.e., \( x_1 - x_2 \), \( y_1 - y_2 \), and \( z_1 - z_2 \).
• Step 2: Square each of these differences.
• Step 3: Add the squared differences.
• Step 4: Take the square root of the sum from Step 3.

This gives us the formula:\[d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}\]
In our specific problem, the distance from point \( P(x, y, z) \) to \( Q(0, 2, 0) \) becomes \( \sqrt{x^2 + (y - 2)^2 + z^2} \). This formula helps determine how far any point \( P \) is from our reference point \( Q \).

A paraboloid is a type of quadric surface. It can appear either as an elliptic or a hyperbolic paraboloid, depending on the coefficients in its equation. In this exercise, we encounter an elliptic paraboloid. This is identified by the equation \( 8y = x^2 + z^2 \).

Here are some characteristics of a paraboloid:

• Its cross-sections parallel to the y-axis are parabolas.
• Its horizontal cross-sections, or those parallel to the x-z plane, are circles or ellipses.
• It typically has a 'bowl' or 'saddle' shape, with its surface curving outward.
• Every point on the surface is equidistant from a fixed line, called the axis.

In our case, the parabolas along the y-axis indicate the points moving away from a central axis, illustrating that distances to a fixed point and a plane can define this surface.

The concept of finding the perpendicular distance from a point to a plane is fundamental when determining if a point lies on a particular plane or how far it is away from it. The formula used is:\[d = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}}\]
where \( Ax + By + Cz + D = 0 \) is the plane equation. The numerator computes the exact "distance along the normal" to the plane, making \( |Ax_1 + By_1 + Cz_1 + D| \) the normal distance.

In the exercise, we use this understanding to calculate how far point \( P(x, y, z) \) is from the plane \( y = -2 \), which simplifies the formula to \( |y + 2| \) since \( y + 2 = 0 \) becomes \( y - (-2) = y + 2 \). Since the coefficients \( A, B, C \) simplify to \( B = 1 \) (taking effect) and \( A = C = 0 \), the denominator reduces to 1, leaving our formula as \( |y + 2| \).

• This form is specifically useful because any point \( P \) that satisfies the condition \( |y + 2| = \sqrt{x^2 + (y - 2)^2 + z^2} \) lies equidistant from both the point \( Q \) and the plane.
• This sets a precise bound  perfect for verifying the surface's formative path.