(I) Find the foot and length of the perpendicular from the point (3,4,5) to the plane :
2 x - 5y + 3z = 39 .
(ii) Find the length and the foot of the perpendicular from the point (7,14,5) to the plane 2x + 4y - z = 2 .

Answer

To solve the problem, we will break it down into two parts as given in the question.

### Part (I): Finding the foot and length of the perpendicular from the point (3, 4, 5) to the plane \(2x - 5y + 3z = 39\).

#### Step 1: Identify the coefficients and the point
The equation of the plane is given as \(2x - 5y + 3z - 39 = 0\). Here, we have:
- \(a = 2\)
- \(b = -5\)
- \(c = 3\)
- \(d = -39\)

The point \(P\) is given as \((x_1, y_1, z_1) = (3, 4, 5)\).

#### Step 2: Calculate the length of the perpendicular
The formula for the length \(L\) of the perpendicular from a point to a plane is given by:
\[ L = \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}} \]

Substituting the values:
\[ L = \frac{|2(3) - 5(4) + 3(5) - 39|}{\sqrt{2^2 + (-5)^2 + 3^2}} \]
Calculating the numerator:
\[ = |6 - 20 + 15 - 39| = |-38| = 38 \]
Calculating the denominator:
\[ = \sqrt{4 + 25 + 9} = \sqrt{38} \]
Thus, the length \(L\) is:
\[ L = \frac{38}{\sqrt{38}} = \sqrt{38} \]

#### Step 3: Find the foot of the perpendicular
The coordinates of the foot of the perpendicular \(Q\) can be found using the formula:
\[ \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} = -\frac{ax_1 + by_1 + cz_1 + d}{a^2 + b^2 + c^2} \]

Calculating the right-hand side:
\[ -\frac{-38}{38} = 1 \]

Thus, we have:
\[ \frac{x - 3}{2} = \frac{y - 4}{-5} = \frac{z - 5}{3} = 1 \]

From this, we can find \(x\), \(y\), and \(z\):
1. \(x - 3 = 2 \implies x = 5\)
2. \(y - 4 = -5 \implies y = -1\)
3. \(z - 5 = 3 \implies z = 8\)

So, the foot of the perpendicular \(Q\) is \((5, -1, 8)\).

### Summary for Part (I):
- Length of the perpendicular: \(\sqrt{38}\)
- Foot of the perpendicular: \((5, -1, 8)\)

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### Part (II): Finding the length and the foot of the perpendicular from the point (7, 14, 5) to the plane \(2x + 4y - z = 2\).

#### Step 1: Identify the coefficients and the point
The equation of the plane is given as \(2x + 4y - z - 2 = 0\). Here, we have:
- \(a = 2\)
- \(b = 4\)
- \(c = -1\)
- \(d = -2\)

The point \(P\) is given as \((x_1, y_1, z_1) = (7, 14, 5)\).

#### Step 2: Calculate the length of the perpendicular
Using the same formula for length \(L\):
\[ L = \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}} \]

Substituting the values:
\[ L = \frac{|2(7) + 4(14) - 5 - 2|}{\sqrt{2^2 + 4^2 + (-1)^2}} \]
Calculating the numerator:
\[ = |14 + 56 - 5 - 2| = |63| = 63 \]
Calculating the denominator:
\[ = \sqrt{4 + 16 + 1} = \sqrt{21} \]
Thus, the length \(L\) is:
\[ L = \frac{63}{\sqrt{21}} = 3\sqrt{21} \]

#### Step 3: Find the foot of the perpendicular
Using the same formula for the coordinates of the foot of the perpendicular \(Q\):
\[ \frac{x - 7}{2} = \frac{y - 14}{4} = \frac{z - 5}{-1} = -\frac{ax_1 + by_1 + cz_1 + d}{a^2 + b^2 + c^2} \]

Calculating the right-hand side:
\[ -\frac{63}{21} = -3 \]

Thus, we have:
\[ \frac{x - 7}{2} = \frac{y - 14}{4} = \frac{z - 5}{-1} = -3 \]

From this, we can find \(x\), \(y\), and \(z\):
1. \(x - 7 = -6 \implies x = 1\)
2. \(y - 14 = -12 \implies y = 2\)
3. \(z - 5 = 3 \implies z = 8\)

So, the foot of the perpendicular \(Q\) is \((1, 2, 8)\).

### Summary for Part (II):
- Length of the perpendicular: \(3\sqrt{21}\)
- Foot of the perpendicular: \((1, 2, 8)\)

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