Find the middle point of the chord intercepted on line `lx + my + n = 0` by circle `x^2+y^2=a^2`.

Options

A. `((-l)/(l^(2)+m^(2)),(-m)/(l^(2)+m^(2)))`
B. `((-ln)/(l^(2)+m^(2)),(-mn)/(l^(2)+m^(2)))`
C. `((-l)/(n(l^(2)+m^(2))),(-m)/(n(l^(2)+m^(2))))`
D. none of these

Explanation

To find the midpoint of the chord intercepted on the line \( lx + my + n = 0 \) by the circle \( x^2 + y^2 = a^2 \), we can follow these steps:

### Step 1: Understand the concept of a chord with a given midpoint
The equation of a chord with a midpoint \( (h, k) \) in a circle \( x^2 + y^2 = a^2 \) can be expressed as:
\[ hx + ky = h^2 + k^2 - a^2 \]

### Step 2: Rewrite the line equation
We have the line equation:
\[ lx + my + n = 0 \]
We can rewrite this as:
\[ lx + my = -n \]

### Step 3: Compare the two equations
Now we will compare the chord equation and the line equation:
1. From the chord equation: \( hx + ky = h^2 + k^2 - a^2 \)
2. From the line equation: \( lx + my = -n \)

### Step 4: Set up the relationships
From the comparison, we can set up the following relationships:
\[ \frac{h}{l} = \frac{k}{m} = \frac{h^2 + k^2 - a^2}{-n} \]

### Step 5: Express \( h \) in terms of \( k \)
From \( \frac{h}{l} = \frac{k}{m} \), we can express \( h \) as:
\[ h = \frac{lk}{m} \]

### Step 6: Substitute \( h \) into the second relationship
Now substitute \( h \) into the second relationship:
\[ \frac{lk/m}{l} = \frac{k}{m} = \frac{(lk/m)^2 + k^2 - a^2}{-n} \]

### Step 7: Simplify the equation
This gives us:
\[ \frac{k}{m} = \frac{(l^2k^2/m^2) + k^2 - a^2}{-n} \]
Multiplying through by \(-n\):
\[ -k n/m = (l^2k^2/m^2) + k^2 - a^2 \]

### Step 8: Rearranging terms
Rearranging gives:
\[ l^2k^2/m^2 + k^2 = -k n/m + a^2 \]

### Step 9: Factor out \( k^2 \)
Factoring out \( k^2 \) gives:
\[ k^2 \left( \frac{l^2}{m^2} + 1 \right) = a^2 - \frac{kn}{m} \]

### Step 10: Solve for \( k \)
Solving for \( k \):
\[ k = -\frac{nm}{l^2 + m^2} \]

### Step 11: Substitute \( k \) back to find \( h \)
Now substitute \( k \) back into the equation for \( h \):
\[ h = \frac{l}{m} \left(-\frac{nm}{l^2 + m^2}\right) = -\frac{ln}{l^2 + m^2} \]

### Final Result
Thus, the coordinates of the midpoint \( (h, k) \) of the chord are:
\[ \left( -\frac{ln}{l^2 + m^2}, -\frac{mn}{l^2 + m^2} \right) \]