If plane passes through the point (1, 1,1) and is perpendicular to the line,`(x-1)/3=(y-1)/0=(z-1)/4`, then its perpendicular distance from the origin is

Options

A. `3/4`
B. `4/3`
C. `7/5`
D. 1

Answer

To find the perpendicular distance from the origin to the plane that passes through the point (1, 1, 1) and is perpendicular to the given line, we can follow these steps:

### Step 1: Identify the direction ratios of the line
The given line is represented as:
\[ \frac{x-1}{3} = \frac{y-1}{0} = \frac{z-1}{4} \]
From this representation, we can extract the direction ratios of the line, which are \(3, 0, 4\).

### Step 2: Determine the normal vector of the plane
Since the plane is perpendicular to the line, the direction ratios of the line will serve as the coefficients of the normal vector of the plane. Thus, the normal vector \(\vec{n}\) of the plane can be represented as:
\[ \vec{n} = (3, 0, 4) \]

### Step 3: Write the equation of the plane
The general equation of a plane can be written as:
\[ a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 \]
where \((x_0, y_0, z_0)\) is a point on the plane and \((a, b, c)\) are the direction ratios of the normal vector.

Substituting the point \((1, 1, 1)\) and the normal vector \((3, 0, 4)\):
\[ 3(x - 1) + 0(y - 1) + 4(z - 1) = 0 \]
Simplifying this, we get:
\[ 3x - 3 + 4z - 4 = 0 \implies 3x + 4z - 7 = 0 \]

### Step 4: Identify coefficients for the distance formula
The equation of the plane is now in the form:
\[ 3x + 0y + 4z - 7 = 0 \]
From this, we can identify:
- \(a = 3\)
- \(b = 0\)
- \(c = 4\)
- \(d = -7\)

### Step 5: Use the distance formula 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:
\[ D = \frac{|ax_0 + by_0 + cz_0 + d|}{\sqrt{a^2 + b^2 + c^2}} \]
Substituting the origin \((0, 0, 0)\) into the formula:
\[ D = \frac{|3(0) + 0(0) + 4(0) - 7|}{\sqrt{3^2 + 0^2 + 4^2}} = \frac{|-7|}{\sqrt{9 + 0 + 16}} = \frac{7}{\sqrt{25}} = \frac{7}{5} \]

### Final Answer
Thus, the perpendicular distance from the origin to the plane is:
\[ \frac{7}{5} \]