The chord of contact of (2,1) w.r.t to the circle `x^(2)+y^(2)+4x+4y+1=0` is

Options

A. `2x+y+7=0`
B. `4x+3y+7=0`
C. `3x+4y+1=0`
D. not existing

Answer

To find the chord of contact of the point (2, 1) with respect to the circle given by the equation \(x^2 + y^2 + 4x + 4y + 1 = 0\), we can follow these steps:

### Step 1: Identify the coefficients from the circle's equation
The general form of a circle's equation is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From the given equation \(x^2 + y^2 + 4x + 4y + 1 = 0\), we can identify:
- \(2g = 4\) which implies \(g = 2\)
- \(2f = 4\) which implies \(f = 2\)
- \(c = 1\)

### Step 2: Write the formula for the chord of contact
The equation of the chord of contact from a point \((x_1, y_1)\) to the circle is given by:
\[ xx_1 + yy_1 + g(x + x_1) + f(y + y_1) + c = 0 \]
Here, \((x_1, y_1) = (2, 1)\).

### Step 3: Substitute the values into the chord of contact formula
Substituting \(x_1 = 2\), \(y_1 = 1\), \(g = 2\), \(f = 2\), and \(c = 1\) into the formula:
\[ x(2) + y(1) + 2(x + 2) + 2(y + 1) + 1 = 0 \]
This simplifies to:
\[ 2x + y + 2x + 4 + 2y + 2 + 1 = 0 \]

### Step 4: Combine like terms
Combining the terms gives:
\[ (2x + 2x) + (y + 2y) + (4 + 2 + 1) = 0 \]
\[ 4x + 3y + 7 = 0 \]

### Step 5: Write the final equation
Thus, the equation of the chord of contact is:
\[ 4x + 3y + 7 = 0 \]

### Final Answer
The chord of contact of the point (2, 1) with respect to the circle is:
\[ 4x + 3y + 7 = 0 \]

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