A variable line moves in such a way that the product of the perpendiculars from (4, 0) and (0, 0) is equal to 9. The locus of the feet of the perpendicular from (0, 0) upon the variable line is a circle, the square of whose radius is

Options

A. 13
B. 15
C. 19
D. 23

Explanation

To solve the given problem step by step, we will follow the outlined process:

### Step 1: Assume the variable line
Let the equation of the variable line be given by:
\[ ax + by + c = 0 \]

### Step 2: Find the foot of the perpendicular
Let the foot of the perpendicular from the origin (0, 0) to the line be denoted as \( (h, k) \). The foot of the perpendicular can be calculated using the formula:
\[ \frac{h}{a} = \frac{k}{b} = -\frac{c}{a^2 + b^2} \]
Let us denote this common value as \( \lambda \):
- \( h = a\lambda \)
- \( k = b\lambda \)
- \( c = -\lambda(a^2 + b^2) \)

### Step 3: Use the condition on the product of perpendiculars
The problem states that the product of the perpendicular distances from the points (4, 0) and (0, 0) to the line is equal to 9.

The formula for the perpendicular distance from a point \( (x_1, y_1) \) to the line \( ax + by + c = 0 \) is given by:
\[ \text{Distance} = \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \]

Calculating the distances:
1. From point \( (4, 0) \):
\[ \text{Distance}_1 = \frac{|4a + c|}{\sqrt{a^2 + b^2}} \]
2. From point \( (0, 0) \):
\[ \text{Distance}_2 = \frac{|c|}{\sqrt{a^2 + b^2}} \]

Thus, the product of the distances is:
\[ \frac{|4a + c|}{\sqrt{a^2 + b^2}} \cdot \frac{|c|}{\sqrt{a^2 + b^2}} = 9 \]
This simplifies to:
\[ \frac{|(4a + c)c|}{a^2 + b^2} = 9 \]

### Step 4: Substitute for \( c \)
Substituting the expression for \( c \):
\[ c = -\lambda(a^2 + b^2) \]
We have:
\[ |4a - \lambda(a^2 + b^2)| \cdot |\lambda(a^2 + b^2)| = 9(a^2 + b^2) \]

### Step 5: Simplify the equation
Let \( D = a^2 + b^2 \). Then the equation becomes:
\[ |4a - \lambda D| \cdot |\lambda D| = 9D \]

### Step 6: Solve for the locus
After simplification, we will arrive at a quadratic equation in terms of \( h \) and \( k \):
\[ h^2 + k^2 - 4h - 9 = 0 \]
This represents a circle.

### Step 7: Identify the center and radius
The standard form of a circle is:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
From our equation \( h^2 + k^2 - 4h - 9 = 0 \), we can complete the square:
\[ (h - 2)^2 + k^2 = 13 \]
This shows that the center of the circle is \( (2, 0) \) and the radius \( r \) is:
\[ r = \sqrt{13} \]

### Step 8: Find the square of the radius
Thus, the square of the radius is:
\[ r^2 = 13 \]

### Final Answer
The square of the radius of the circle is:
\[ \boxed{13} \]