on the surface
z=2⋅y^2+2⋅x^2
the closest point to the plain: z=y+3⋅x−3
is: (4/1 , 1/4 , 5/4)
what is the distance?

To find the distance between the given point \((4, 1/4, 5/4)\) and the plane \( z = y + 3x - 3 \), we use the distance formula from a point to a plane.

The general formula for the distance \( D \) from a point \( (x_1, y_1, z_1) \) to a plane \( Ax + By + Cz + D = 0 \) is:

\[ D = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} \]

First, rewrite the plane equation \( z = y + 3x - 3 \) in the form \( Ax + By + Cz + D = 0 \):

\[ z - y - 3x + 3 = 0 \]

Here, \( A = -3 \), \( B = -1 \), \( C = 1 \), and \( D = 3 \).

Now, plug the point \( (4, \frac{1}{4}, \frac{5}{4}) \) into the formula:

\[ D = \frac{|-3(4) - 1\left(\frac{1}{4}\right) + 1\left(\frac{5}{4}\right) + 3|}{\sqrt{(-3)^2 + (-1)^2 + 1^2}} \]

Calculate the numerator:

\[ -3(4) = -12 \]
\[ -1\left(\frac{1}{4}\right) = -\frac{1}{4} \]
\[ 1\left(\frac{5}{4}\right) = \frac{5}{4} \]
\[ -12 - \frac{1}{4} + \frac{5}{4} + 3 \]

Combine the terms:

\[ -12 - \frac{1}{4} + \frac{5}{4} + 3 = -12 - \frac{1}{4} + \frac{5}{4} + 3 \]
\[ = -12 + \frac{4}{4} \cdot 3 - \frac{1}{4} + \frac{5}{4} \]
\[ = -12 + 3 - \frac{1}{4} + \frac{5}{4} \]
\[ = -12 + 3 + \frac{4}{4} = -12 + 3 + 1 \]
\[ = -12 + 4 \]
\[ = -8 + 1 = -9 \]

The absolute value of the numerator is:

\[ |-8| = 8 \]

Now, calculate the denominator:

\[ \sqrt{(-3)^2 + (-1)^2 + 1^2} = \sqrt{9 + 1 + 1} = \sqrt{11} \]

Finally, compute the distance \( D \):

\[ D = \frac{8}{\sqrt{11}} \]

For the distance in a more standard form, rationalize the denominator:

\[ D = \frac{8}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}} = \frac{8\sqrt{11}}{11} \]

So, the distance between the point \((4, 1/4, 5/4)\) and the plane \( z = y + 3x - 3 \) is:

\[ D = \frac{8\sqrt{11}}{11} \]

Do you want more details or have any questions? Here are some additional questions you might consider:

1. How do we derive the formula for the distance from a point to a plane?
2. What is the significance of rationalizing the denominator in our final answer?
3. How do we verify if a point lies on a given plane?
4. What are the applications of finding the distance from a point to a plane in real-world problems?
5. How would the distance calculation change if the plane equation were in a different form?
6. Can the method be generalized for higher-dimensional spaces?
7. How can we find the coordinates of the point on the plane closest to the given point?
8. How does the concept of orthogonal projection relate to this problem?

**Tip:** Always check if the point lies on the plane before calculating the distance; if it does, the distance is zero.

Explore how to calculate the distance between a point and a plane in analytic geometry. Step-by-step solution provided for the problem involving the point (4, 1/4, 5/4) and the plane z = y + 3x - 3. Suitable for undergraduate level students studying analytic geometry and plane equations.

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