The equation of axis of the parabola having focus (2,3) and directrix `x-4y+3=0` is

Options

A. `x-4y-11=0`
B. `4x-y-11=0`
C. `x+4y+11=0`
D. `4x+y-11=0`

Answer

To find the equation of the axis of the parabola with a given focus and directrix, we can follow these steps:

### Step 1: Identify the Focus and Directrix
The focus of the parabola is given as \( S(2, 3) \) and the directrix is given by the equation:
\[ x - 4y + 3 = 0 \]

### Step 2: Determine the Slope of the Directrix
Rearranging the directrix equation into slope-intercept form \( y = mx + b \):
\[ x - 4y + 3 = 0 \implies 4y = x + 3 \implies y = \frac{1}{4}x + \frac{3}{4} \]
From this, we can see that the slope \( m_1 \) of the directrix is \( \frac{1}{4} \).

### Step 3: Find the Slope of the Axis
The axis of the parabola is perpendicular to the directrix. If two lines are perpendicular, the product of their slopes is \( -1 \). Therefore, if \( m_1 \) is the slope of the directrix, the slope \( m_2 \) of the axis can be calculated as:
\[ m_1 \cdot m_2 = -1 \implies \frac{1}{4} \cdot m_2 = -1 \implies m_2 = -4 \]

### Step 4: Write the Equation of the Axis
The axis of the parabola will pass through the focus \( S(2, 3) \) and have a slope of \( -4 \). We can use the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]
Substituting \( (x_1, y_1) = (2, 3) \) and \( m = -4 \):
\[ y - 3 = -4(x - 2) \]

### Step 5: Simplify the Equation
Expanding and rearranging the equation:
\[ y - 3 = -4x + 8 \implies y = -4x + 11 \]
Rearranging to standard form:
\[ 4x + y - 11 = 0 \]

### Final Answer
Thus, the equation of the axis of the parabola is:
\[ \boxed{4x + y - 11 = 0} \]

---