Locus of midpoints of chords of circles `x^(2)+y^(2)-4x-2y-4=0` which are perpendicular to the line `4x-3y+10=0` is

Options

A. `4x-3y+5=0`
B. `4x-3y-5=0`
C. `4x-3y+10=0`
D. `4x+3y+5=0`

Answer

To find the locus of the midpoints of chords of the circle given by the equation \( x^2 + y^2 - 4x - 2y - 4 = 0 \) that are perpendicular to the line \( 4x - 3y + 10 = 0 \), we will follow these steps:

### Step 1: Rewrite the Circle's Equation
First, we will rewrite the equation of the circle in standard form. The given equation is:

\[ x^2 + y^2 - 4x - 2y - 4 = 0 \]

We can complete the square for both \(x\) and \(y\):

1. For \(x\):
\[ x^2 - 4x = (x - 2)^2 - 4 \]

2. For \(y\):
\[ y^2 - 2y = (y - 1)^2 - 1 \]

Substituting these back into the equation gives:

\[ (x - 2)^2 - 4 + (y - 1)^2 - 1 - 4 = 0 \]

Simplifying this, we get:

\[ (x - 2)^2 + (y - 1)^2 = 9 \]

This represents a circle centered at \( (2, 1) \) with a radius of \( 3 \).

### Step 2: Determine the Slope of the Given Line
Next, we need to find the slope of the line \( 4x - 3y + 10 = 0 \). We can rewrite this in slope-intercept form:

\[ 3y = 4x + 10 \implies y = \frac{4}{3}x + \frac{10}{3} \]

Thus, the slope \( m_2 \) of this line is \( \frac{4}{3} \).

### Step 3: Find the Slope of the Chords
Let the midpoint of the chord be \( P(h, k) \). The slope \( m_1 \) of the chord that is perpendicular to the line will be:

\[ m_1 = -\frac{1}{m_2} = -\frac{3}{4} \]

### Step 4: Use the Midpoint Formula
The equation of the chord with midpoint \( P(h, k) \) can be expressed using the formula for the chord of a circle:

\[ T = S_1 \]

Where \( T \) is the equation of the chord and \( S_1 \) is the equation of the circle evaluated at \( (h, k) \).

The equation of the chord is:

\[ h(x - h) + k(y - k) = 9 \]

Substituting the values gives:

\[ hx + ky - 9 - h^2 - k^2 = 0 \]

### Step 5: Set Up the Perpendicular Condition
Since the slope of the chord is \( -\frac{3}{4} \), we can express this as:

\[ \frac{k - y_1}{h - x_1} = -\frac{3}{4} \]

This leads to the equation:

\[ 4(k - y_1) + 3(h - x_1) = 0 \]

### Step 6: Solve for the Locus
From the perpendicularity condition, we can derive:

\[ 4h + 3k - 5 = 0 \]

This is the equation of the locus of midpoints of the chords.

### Final Step: Replace Variables
Finally, we replace \( h \) and \( k \) with \( x \) and \( y \):

\[ 4x - 3y - 5 = 0 \]

### Conclusion
Thus, the locus of the midpoints of the chords of the circle that are perpendicular to the given line is:

\[ \boxed{4x - 3y - 5 = 0} \]