In the following, find the distance of each of the given points from the corresponding given planes :
`{:("Point","Plane"),("(i) (0,0,0)",2x - y + 2z +1 = 0 ),((ii) (3,-2,1),2x-y + 2z + 3 = 0 ),((iii)(-6, 0,0),2x-3y + 6z-2 = 0 ),((iv) (2,3,-5),x + 2y - 2z = 9 .):}`

Answer

To find the distance of each given point from the corresponding plane, we can use the formula for the distance \( D \) from a point \( (x_1, y_1, z_1) \) to the plane given by the equation \( ax + by + cz + d = 0 \):

\[ D = \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}} \]

where \( (a, b, c) \) are the coefficients of \( x, y, z \) in the plane equation, and \( d \) is the constant term.

Let's solve each part step by step:

### (i) Point: \( (0, 0, 0) \), Plane: \( 2x - y + 2z + 1 = 0 \)

1. Identify coefficients: \( a = 2, b = -1, c = 2, d = 1 \).
2. Substitute point coordinates into the distance formula:
\[ D = \frac{|2(0) + (-1)(0) + 2(0) + 1|}{\sqrt{2^2 + (-1)^2 + 2^2}} = \frac{|1|}{\sqrt{4 + 1 + 4}} = \frac{1}{\sqrt{9}} = \frac{1}{3} \]

### (ii) Point: \( (3, -2, 1) \), Plane: \( 2x - y + 2z + 3 = 0 \)

1. Identify coefficients: \( a = 2, b = -1, c = 2, d = 3 \).
2. Substitute point coordinates into the distance formula:
\[ D = \frac{|2(3) + (-1)(-2) + 2(1) + 3|}{\sqrt{2^2 + (-1)^2 + 2^2}} = \frac{|6 + 2 + 2 + 3|}{\sqrt{4 + 1 + 4}} = \frac{|13|}{\sqrt{9}} = \frac{13}{3} \]

### (iii) Point: \( (-6, 0, 0) \), Plane: \( 2x - 3y + 6z - 2 = 0 \)

1. Identify coefficients: \( a = 2, b = -3, c = 6, d = -2 \).
2. Substitute point coordinates into the distance formula:
\[ D = \frac{|2(-6) + (-3)(0) + 6(0) - 2|}{\sqrt{2^2 + (-3)^2 + 6^2}} = \frac{|-12 - 2|}{\sqrt{4 + 9 + 36}} = \frac{14}{\sqrt{49}} = \frac{14}{7} = 2 \]

### (iv) Point: \( (2, 3, -5) \), Plane: \( x + 2y - 2z = 9 \)

1. Rewrite the plane equation: \( x + 2y - 2z - 9 = 0 \) (thus \( d = -9 \)).
2. Identify coefficients: \( a = 1, b = 2, c = -2, d = -9 \).
3. Substitute point coordinates into the distance formula:
\[ D = \frac{|1(2) + 2(3) - 2(-5) - 9|}{\sqrt{1^2 + 2^2 + (-2)^2}} = \frac{|2 + 6 + 10 - 9|}{\sqrt{1 + 4 + 4}} = \frac{|9|}{\sqrt{9}} = \frac{9}{3} = 3 \]

### Summary of Distances:
1. Distance from \( (0, 0, 0) \) to the plane: \( \frac{1}{3} \)
2. Distance from \( (3, -2, 1) \) to the plane: \( \frac{13}{3} \)
3. Distance from \( (-6, 0, 0) \) to the plane: \( 2 \)
4. Distance from \( (2, 3, -5) \) to the plane: \( 3 \)