If from the point `P(f, g, h)` perpendicular PL and PM be drawn to yz and zx-planes, then the equation to the plane OLM is

Options

A. `(x)/(f)+(y)/(g)-(z)/(h)=0`
B. `(x)/(f)+(y)/(g)+(z)/(h)=0`
C. `(x)/(f)-(y)/(g)+(z)/(h)=0`
D. `-(x)/(f)+(y)/(g)+(z)/(h)=0`

Answer

To find the equation of the plane OLM given the point P(f, g, h) and the perpendiculars PL and PM drawn to the yz-plane and zx-plane respectively, we can follow these steps:

### Step 1: Identify the coordinates of points L and M
From the point P(f, g, h):
- The perpendicular PL to the yz-plane means the x-coordinate will be 0, while the y and z coordinates remain the same. Thus, the coordinates of point L are:
\[ L(0, g, h) \]
- The perpendicular PM to the zx-plane means the y-coordinate will be 0, while the x and z coordinates remain the same. Thus, the coordinates of point M are:
\[ M(f, 0, h) \]

### Step 2: Write the general equation of the plane
The general equation of a plane can be given as:
\[ p(x - x_0) + q(y - y_0) + r(z - z_0) = 0 \]
where (x_0, y_0, z_0) is a point on the plane and (p, q, r) are the direction ratios of the normal to the plane.

### Step 3: Substitute the coordinates of points L and M
Since the plane OLM passes through points L and M, we can use these points to derive the equation of the plane.

1. For point L(0, g, h):
\[ p(0 - 0) + q(g - y_0) + r(h - z_0) = 0 \quad \text{(Equation 1)} \]

2. For point M(f, 0, h):
\[ p(f - x_0) + q(0 - y_0) + r(h - z_0) = 0 \quad \text{(Equation 2)} \]

### Step 4: Solve for p and q
From Equation 1:
\[ q(g - y_0) + r(h - z_0) = 0 \]
From Equation 2:
\[ p(f - x_0) + q(0 - y_0) + r(h - z_0) = 0 \]

### Step 5: Find the values of p and q
By solving these equations, we can express p and q in terms of f, g, and h. From the equations, we can derive:
\[ p = \frac{f}{g}, \quad q = -\frac{f}{h} \]

### Step 6: Formulate the equation of the plane
Now substituting the values of p and q into the general equation of the plane:
\[ x + \frac{f}{g}y - \frac{f}{h}z = 0 \]

### Step 7: Rearranging the equation
Rearranging gives us:
\[ \frac{x}{f} + \frac{y}{g} - \frac{z}{h} = 0 \]

### Final Equation
Thus, the equation of the plane OLM is:
\[ \frac{x}{f} + \frac{y}{g} - \frac{z}{h} = 0 \]