The product of the perpendiculars from the two points `(pm 4 , 0)` to the line 3x `cos phi + 5y sin phi = 15` is

Options

A. 25
B. 16
C. 9
D. 8

Answer

To solve the problem of finding the product of the perpendiculars from the points \( (4, 0) \) and \( (-4, 0) \) to the line given by the equation \( 3x \cos \phi + 5y \sin \phi = 15 \), we can follow these steps:

### Step 1: Identify the line equation and points
The line equation is given as:
\[ 3x \cos \phi + 5y \sin \phi = 15 \]
The two points from which we need to find the perpendicular distances are:
\[ P_1(4, 0) \quad \text{and} \quad P_2(-4, 0) \]

### Step 2: Calculate the perpendicular distance from point \( P_1(4, 0) \)
The formula for the perpendicular distance \( d \) from a point \( (x_0, y_0) \) to the line \( Ax + By + C = 0 \) is given by:
\[ d = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]
Rearranging the line equation, we can express it in the standard form:
\[ 3x \cos \phi + 5y \sin \phi - 15 = 0 \]
Here, \( A = 3 \cos \phi \), \( B = 5 \sin \phi \), and \( C = -15 \).

Now, substituting \( P_1(4, 0) \):
\[ d_1 = \frac{|3(4) \cos \phi + 5(0) \sin \phi - 15|}{\sqrt{(3 \cos \phi)^2 + (5 \sin \phi)^2}} \]
Calculating \( d_1 \):
\[ d_1 = \frac{|12 \cos \phi - 15|}{\sqrt{9 \cos^2 \phi + 25 \sin^2 \phi}} \]

### Step 3: Calculate the perpendicular distance from point \( P_2(-4, 0) \)
Now, substituting \( P_2(-4, 0) \):
\[ d_2 = \frac{|3(-4) \cos \phi + 5(0) \sin \phi - 15|}{\sqrt{(3 \cos \phi)^2 + (5 \sin \phi)^2}} \]
Calculating \( d_2 \):
\[ d_2 = \frac{|-12 \cos \phi - 15|}{\sqrt{9 \cos^2 \phi + 25 \sin^2 \phi}} \]

### Step 4: Calculate the product of the distances
The product of the distances \( P_1 \) and \( P_2 \) is:
\[ P_1 \cdot P_2 = d_1 \cdot d_2 = \left(\frac{|12 \cos \phi - 15|}{\sqrt{9 \cos^2 \phi + 25 \sin^2 \phi}}\right) \cdot \left(\frac{|-12 \cos \phi - 15|}{\sqrt{9 \cos^2 \phi + 25 \sin^2 \phi}}\right) \]
This simplifies to:
\[ P_1 \cdot P_2 = \frac{|(12 \cos \phi - 15)(-12 \cos \phi - 15)|}{9 \cos^2 \phi + 25 \sin^2 \phi} \]

### Step 5: Simplify the expression
Expanding the numerator:
\[ (12 \cos \phi - 15)(-12 \cos \phi - 15) = -144 \cos^2 \phi - 15(12 \cos \phi) + 15(12 \cos \phi) + 225 = 144 \cos^2 \phi - 225 \]
Thus, we have:
\[ P_1 \cdot P_2 = \frac{144 \cos^2 \phi - 225}{9 \cos^2 \phi + 25 \sin^2 \phi} \]

### Step 6: Final result
The final expression for the product of the perpendiculars is:
\[ P_1 \cdot P_2 = \frac{144 \cos^2 \phi - 225}{9 \cos^2 \phi + 25 \sin^2 \phi} \]