If a plane, intercepts on the coordinates axes at 8,4,4 then the length of the perpendicular from the origin to the plane is

Options

A. `8/3` units
B. `3/8 `units
C. 3 units
D. `4/5` units

Answer

To find the length of the perpendicular from the origin to the plane that intercepts the coordinate axes at (8, 4, 4), we can follow these steps:

### Step 1: Identify the intercepts
The intercepts of the plane on the x, y, and z axes are given as:
- x-intercept (a) = 8
- y-intercept (b) = 4
- z-intercept (c) = 4

### Step 2: Write the equation of the plane
The general equation of a plane that intercepts the axes at (a, b, c) is given by:
\[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \]
Substituting the values of a, b, and c:
\[ \frac{x}{8} + \frac{y}{4} + \frac{z}{4} = 1 \]

### Step 3: Clear the fractions
To eliminate the fractions, multiply the entire equation by the least common multiple (LCM) of the denominators (which is 8):
\[ x + 2y + 2z = 8 \]
Rearranging gives us the standard form of the equation of the plane:
\[ x + 2y + 2z - 8 = 0 \]

### Step 4: Use the distance formula
To find the distance \(d\) from the origin (0, 0, 0) to the plane given by the equation \(Ax + By + Cz + D = 0\), we use the formula:
\[ d = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}} \]
where \((x_0, y_0, z_0)\) is the point from which we are measuring the distance (in this case, the origin).

### Step 5: Substitute the values into the formula
Here, \(A = 1\), \(B = 2\), \(C = 2\), and \(D = -8\). Substituting the coordinates of the origin (0, 0, 0):
\[ d = \frac{|1(0) + 2(0) + 2(0) - 8|}{\sqrt{1^2 + 2^2 + 2^2}} \]
This simplifies to:
\[ d = \frac{|-8|}{\sqrt{1 + 4 + 4}} = \frac{8}{\sqrt{9}} = \frac{8}{3} \]

### Final Answer
The length of the perpendicular from the origin to the plane is:
\[ \frac{8}{3} \text{ units} \]

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