The normal of the circle `(x- 2)^(2)+ (y- 1)^(2) =16` which bisects the chord cut off by the line x-2y-3=0 is

Options

A. `2x+y+3=0`
B. `2x+y-4=0`
C. `2x+y-5=0`
D. `2x+y-7=0`

Answer

To solve the problem, we need to find the equation of the normal to the given circle that bisects the chord cut off by the given line. Let's go through the steps systematically.

### Step 1: Identify the center of the circle
The equation of the circle is given as:
\[ (x - 2)^2 + (y - 1)^2 = 16 \]
From this equation, we can identify the center of the circle, which is at the point \((h, k) = (2, 1)\).

**Hint:** The center of a circle in the standard form \((x - h)^2 + (y - k)^2 = r^2\) is \((h, k)\).

### Step 2: Find the slope of the normal
Let the slope of the normal be \(m\). The equation of the normal line at the center of the circle can be expressed as:
\[ y - 1 = m(x - 2) \]
This can be rearranged to:
\[ y = mx - 2m + 1 \]

**Hint:** The equation of a line in point-slope form is \(y - y_1 = m(x - x_1)\).

### Step 3: Determine the slope of the given line
The line given is:
\[ x - 2y - 3 = 0 \]
We can rearrange this to slope-intercept form:
\[ 2y = x - 3 \implies y = \frac{1}{2}x - \frac{3}{2} \]
From this, we see that the slope of the line is \(\frac{1}{2}\).

**Hint:** To find the slope from the equation \(Ax + By + C = 0\), rearrange it to \(y = mx + b\).

### Step 4: Use the property of perpendicular lines
Since the normal bisects the chord cut off by the line, the normal and the line are perpendicular. Therefore, the product of their slopes must equal \(-1\):
\[ m \cdot \frac{1}{2} = -1 \]
Solving for \(m\):
\[ m = -2 \]

**Hint:** For two perpendicular lines with slopes \(m_1\) and \(m_2\), the relationship is \(m_1 \cdot m_2 = -1\).

### Step 5: Substitute \(m\) back into the normal equation
Now substituting \(m = -2\) into the equation of the normal:
\[ y = -2x - 2(-2) + 1 \]
This simplifies to:
\[ y = -2x + 4 + 1 \implies y = -2x + 5 \]

**Hint:** Simplifying an equation often involves combining like terms.

### Step 6: Convert to standard form
To express the equation in standard form \(Ax + By + C = 0\):
\[ 2x + y - 5 = 0 \]

**Hint:** Rearranging to standard form involves moving all terms to one side of the equation.

### Final Answer
The equation of the normal to the circle that bisects the chord cut off by the line is:
\[ 2x + y - 5 = 0 \]