The vertex of parabola whose focus is `(2,1)` and directrix is x - 2y + 10 = 0 is

Options

A. a) (2,2)
B. b) (1,3)
C. c) (3,1)
D. d) (0,5)

Answer

To find the vertex of the parabola with a focus at \( (2, 1) \) and a directrix given by the equation \( x - 2y + 10 = 0 \), we can follow these steps:

### Step 1: Identify the focus and the directrix
The focus \( F \) is given as \( (2, 1) \). The directrix is given by the equation \( x - 2y + 10 = 0 \).

### Step 2: Find the coefficients of the directrix
Rearranging the directrix equation into the standard form \( Ax + By + C = 0 \):
- Here, \( A = 1 \), \( B = -2 \), and \( C = 10 \).

### Step 3: Find the foot of the perpendicular from the focus to the directrix
The foot of the perpendicular \( M(h, k) \) from the point \( F(2, 1) \) to the line can be found using the formula:

\[ \frac{h - x_1}{A} = \frac{k - y_1}{B} = \frac{-Ax_1 + By_1 + C}{A^2 + B^2} \]

Substituting \( (x_1, y_1) = (2, 1) \):
\[ \frac{h - 2}{1} = \frac{k - 1}{-2} = \frac{-1 \cdot 2 + (-2) \cdot 1 + 10}{1^2 + (-2)^2} \]

### Step 4: Calculate the right-hand side
Calculating the right-hand side:
\[ = \frac{-2 - 2 + 10}{1 + 4} = \frac{6}{5} \]

### Step 5: Set up the equations
Now we have:
\[ h - 2 = \frac{6}{5} \quad \text{and} \quad k - 1 = -2 \cdot \frac{6}{5} \]

From the first equation:
\[ h = 2 + \frac{6}{5} = \frac{10}{5} + \frac{6}{5} = \frac{16}{5} \]

From the second equation:
\[ k = 1 - \frac{12}{5} = \frac{5}{5} - \frac{12}{5} = -\frac{7}{5} \]

So, the coordinates of point \( M \) are \( \left(\frac{16}{5}, -\frac{7}{5}\right) \).

### Step 6: Find the vertex \( V \)
The vertex \( V \) is the midpoint of \( F(2, 1) \) and \( M\left(\frac{16}{5}, -\frac{7}{5}\right) \):
\[ V = \left(\frac{2 + \frac{16}{5}}{2}, \frac{1 - \frac{7}{5}}{2}\right) \]

Calculating the x-coordinate:
\[ \frac{2 + \frac{16}{5}}{2} = \frac{\frac{10}{5} + \frac{16}{5}}{2} = \frac{\frac{26}{5}}{2} = \frac{26}{10} = \frac{13}{5} \]

Calculating the y-coordinate:
\[ \frac{1 - \frac{7}{5}}{2} = \frac{\frac{5}{5} - \frac{7}{5}}{2} = \frac{-\frac{2}{5}}{2} = -\frac{2}{10} = -\frac{1}{5} \]

### Final Answer
Thus, the vertex of the parabola is:
\[ V = \left(\frac{13}{5}, -\frac{1}{5}\right) \]