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

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

The equation of the plane is:
\[ z = y + 3x - 3 \]
We rewrite it in the general form \(Ax + By + Cz + D = 0\):
\[ z - y - 3x + 3 = 0 \]

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

The formula for the distance \(D\) from a point \((x_1, y_1, z_1)\) to a plane \(Ax + By + Cz + D = 0\) is:
\[ D = \frac{|Ax_1 + By_1 + Cz_1 + D|}{\sqrt{A^2 + B^2 + C^2}} \]

Plugging in the point \((3/4, 1/4, 5/4)\):
\[ D = \frac{|-3 \cdot \frac{3}{4} - 1 \cdot \frac{1}{4} + 1 \cdot \frac{5}{4} + 3|}{\sqrt{(-3)^2 + (-1)^2 + 1^2}} \]

Calculate the numerator:
\[ -3 \cdot \frac{3}{4} = -\frac{9}{4} \]
\[ -1 \cdot \frac{1}{4} = -\frac{1}{4} \]
\[ 1 \cdot \frac{5}{4} = \frac{5}{4} \]
\[ \text{Sum these values:} \]
\[ -\frac{9}{4} - \frac{1}{4} + \frac{5}{4} + 3 \]

Combine the terms:
\[ -\frac{9}{4} - \frac{1}{4} + \frac{5}{4} + \frac{12}{4} = -\frac{10}{4} + \frac{5}{4} + \frac{12}{4} = -\frac{10}{4} + \frac{17}{4} = \frac{7}{4} \]

The absolute value of the numerator is:
\[ \left|\frac{7}{4}\right| = \frac{7}{4} \]

Now, calculate the denominator:
\[ \sqrt{(-3)^2 + (-1)^2 + 1^2} = \sqrt{9 + 1 + 1} = \sqrt{11} \]

Finally, compute the distance \(D\):
\[ D = \frac{\frac{7}{4}}{\sqrt{11}} = \frac{7}{4\sqrt{11}} \]

To rationalize the denominator:
\[ D = \frac{7}{4\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}} = \frac{7\sqrt{11}}{44} \]

So, the distance between the point \((3/4, 1/4, 5/4)\) and the plane \(z = y + 3x - 3\) is:
\[ D = \frac{7\sqrt{11}}{44} \]

Would you like more details or have any questions? Here are some additional questions to 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 double-check the algebraic manipulations when dealing with fractions and square roots to ensure accuracy in your final result.

Learn how to calculate the distance from the point (3/4, 1/4, 5/4) to the plane z = y + 3x - 3 with a detailed step-by-step solution. This problem involves finding the distance using the distance formula and understanding coordinate geometry concepts. Suitable for advanced high school students.

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