The equation of the circle whose centre lies on `x+2y=0` and touching 3x-4y+8=0 and 3x-4y-28=0 is

Options

A. `(x-2)^2+(y+1)^(2)=324`
B. `5(x-2)^(2)+5(y-2)^(2)=324`
C. `25(x-2)^(2)+25(y+1)^(2)=324`
D. none

Answer

To find the equation of the circle whose center lies on the line \( x + 2y = 0 \) and touches the lines \( 3x - 4y + 8 = 0 \) and \( 3x - 4y - 28 = 0 \), we can follow these steps:

### Step 1: Identify the distance between the two parallel lines
The two lines given are:
1. \( 3x - 4y + 8 = 0 \) (let's call this Line 1)
2. \( 3x - 4y - 28 = 0 \) (let's call this Line 2)

Since both lines have the same coefficients for \( x \) and \( y \), they are parallel. The distance \( d \) between two parallel lines \( Ax + By + C_1 = 0 \) and \( Ax + By + C_2 = 0 \) is given by the formula:

\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]

For our lines:
- \( A = 3 \)
- \( B = -4 \)
- \( C_1 = 8 \)
- \( C_2 = -28 \)

Substituting these values into the formula:

\[ d = \frac{|-28 - 8|}{\sqrt{3^2 + (-4)^2}} = \frac{|-36|}{\sqrt{9 + 16}} = \frac{36}{5} = 7.2 \]

### Step 2: Find the radius of the circle
Since the circle touches both lines, the distance between the two lines equals the diameter of the circle. Therefore, the radius \( r \) of the circle is half of the distance:

\[ r = \frac{d}{2} = \frac{36/5}{2} = \frac{18}{5} \]

### Step 3: Determine the center of the circle
The center of the circle lies on the line \( x + 2y = 0 \). We can express the center as \( (h, k) \) where \( k = -\frac{h}{2} \).

### Step 4: Use the distance formula to find the center
The distance from the center \( (h, k) \) to Line 1 must equal the radius \( r \). The distance \( D \) from a point \( (x_0, y_0) \) to a line \( Ax + By + C = 0 \) is given by:

\[ D = \frac{|Ax_0 + By_0 + C|}{\sqrt{A^2 + B^2}} \]

For Line 1 \( 3x - 4y + 8 = 0 \):
- \( A = 3 \)
- \( B = -4 \)
- \( C = 8 \)

Substituting \( (h, -\frac{h}{2}) \):

\[ D = \frac{|3h - 4(-\frac{h}{2}) + 8|}{\sqrt{3^2 + (-4)^2}} = \frac{|3h + 2h + 8|}{5} = \frac{|5h + 8|}{5} \]

Setting this equal to the radius \( \frac{18}{5} \):

\[ \frac{|5h + 8|}{5} = \frac{18}{5} \]

Multiplying both sides by 5:

\[ |5h + 8| = 18 \]

This gives us two equations:

1. \( 5h + 8 = 18 \)
2. \( 5h + 8 = -18 \)

Solving the first equation:

\[ 5h = 10 \implies h = 2 \]

Solving the second equation:

\[ 5h = -26 \implies h = -\frac{26}{5} \]

### Step 5: Find corresponding \( k \) values
For \( h = 2 \):

\[ k = -\frac{2}{2} = -1 \]

For \( h = -\frac{26}{5} \):

\[ k = -\frac{-\frac{26}{5}}{2} = \frac{13}{5} \]

### Step 6: Write the equations of the circles
The general equation of a circle is given by:

\[ (x - h)^2 + (y - k)^2 = r^2 \]

1. For \( (h, k) = (2, -1) \) and \( r = \frac{18}{5} \):

\[ (x - 2)^2 + (y + 1)^2 = \left(\frac{18}{5}\right)^2 \]

Calculating \( r^2 \):

\[ r^2 = \frac{324}{25} \]

So the equation becomes:

\[ (x - 2)^2 + (y + 1)^2 = \frac{324}{25} \]

2. For \( (h, k) = \left(-\frac{26}{5}, \frac{13}{5}\right) \):

\[ \left(x + \frac{26}{5}\right)^2 + \left(y - \frac{13}{5}\right)^2 = \frac{324}{25} \]

### Final Answer
The equation of the circle can be expressed as:

\[ 25(x - 2)^2 + 25(y + 1)^2 = 324 \]