perpendicular distance of the origin from the plane which makes intercepts `12,6 and 4` on X,Y,Z axes respectively, is :

Options

A. 13
B. 11
C. 17
D. None of these

Answer

To find the perpendicular distance of the origin from the plane that makes intercepts 12, 6, and 4 on the X, Y, and Z axes respectively, we can follow these steps:

### Step 1: Write the equation of the plane in intercept form
The intercept form of a plane is given by the equation:

\[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \]

where \(a\), \(b\), and \(c\) are the intercepts on the X, Y, and Z axes respectively. Here, we have \(a = 12\), \(b = 6\), and \(c = 4\).

### Step 2: Substitute the values into the equation
Substituting the values of \(a\), \(b\), and \(c\) into the intercept form, we get:

\[ \frac{x}{12} + \frac{y}{6} + \frac{z}{4} = 1 \]

### Step 3: Clear the denominators
To eliminate the fractions, we can multiply through by the least common multiple (LCM) of the denominators, which is 12:

\[ 12 \left(\frac{x}{12} + \frac{y}{6} + \frac{z}{4}\right) = 12 \]

This simplifies to:

\[ x + 2y + 3z = 12 \]

### Step 4: Rearrange the equation into standard form
Rearranging this equation gives us the standard form of the plane:

\[ x + 2y + 3z - 12 = 0 \]

### Step 5: Use the formula for the perpendicular distance from a point to a plane
The formula for the perpendicular distance \(D\) from a point \((x_0, y_0, z_0)\) to the plane \(Ax + By + Cz + D = 0\) is given by:

\[ D = \frac{|Ax_0 + By_0 + Cz_0 + D|}{\sqrt{A^2 + B^2 + C^2}} \]

In our case, the coefficients are:
- \(A = 1\)
- \(B = 2\)
- \(C = 3\)
- \(D = -12\)

And the point is the origin \((0, 0, 0)\).

### Step 6: Substitute the values into the distance formula
Substituting the values into the formula:

\[ D = \frac{|1(0) + 2(0) + 3(0) - 12|}{\sqrt{1^2 + 2^2 + 3^2}} \]

This simplifies to:

\[ D = \frac{|-12|}{\sqrt{1 + 4 + 9}} = \frac{12}{\sqrt{14}} \]

### Final Result:
Thus, the perpendicular distance of the origin from the plane is:

\[ D = \frac{12}{\sqrt{14}} \text{ units} \]