If x+2y=3 and 2x+y=3 touches a circle whose centres lies on 3x+4y=5, then possible co-ordinates of centres of circle are

Options

A. `(4/3,1/4)`
B. `(-3,2)`
C. `(5/7,5/7)`
D. `(3,-1)`

Answer

To find the possible coordinates of the center of the circle that touches the lines \(x + 2y = 3\) and \(2x + y = 3\), and whose center lies on the line \(3x + 4y = 5\), we can follow these steps:

### Step 1: Identify the equations of the lines
We have two lines:
1. \(L_1: x + 2y = 3\)
2. \(L_2: 2x + y = 3\)

### Step 2: Find the point of intersection of the lines
To find the point of intersection, we can solve the two equations simultaneously.

From \(L_1\):
\[ x + 2y = 3 \]
\[ x = 3 - 2y \]

Substituting \(x\) in \(L_2\):
\[ 2(3 - 2y) + y = 3 \]
\[ 6 - 4y + y = 3 \]
\[ 6 - 3y = 3 \]
\[ 3y = 3 \]
\[ y = 1 \]

Now substituting \(y = 1\) back into \(L_1\):
\[ x + 2(1) = 3 \]
\[ x + 2 = 3 \]
\[ x = 1 \]

So, the point of intersection is \((1, 1)\).

### Step 3: Find the distance from the center to the lines
Let the center of the circle be \((h, k)\). The distance from the center to the line \(L_1\) is given by:
\[ d_1 = \frac{|h + 2k - 3|}{\sqrt{1^2 + 2^2}} = \frac{|h + 2k - 3|}{\sqrt{5}} \]

The distance from the center to the line \(L_2\) is given by:
\[ d_2 = \frac{|2h + k - 3|}{\sqrt{2^2 + 1^2}} = \frac{|2h + k - 3|}{\sqrt{5}} \]

### Step 4: Set the distances equal to the radius
Since the circle touches both lines, the distances \(d_1\) and \(d_2\) must be equal:
\[ |h + 2k - 3| = |2h + k - 3| \]

### Step 5: Solve the absolute value equation
This gives us two cases to consider:

**Case 1:**
\[ h + 2k - 3 = 2h + k - 3 \]
Simplifying this:
\[ h + 2k = 2h + k \]
\[ k = h \]

**Case 2:**
\[ h + 2k - 3 = -(2h + k - 3) \]
Simplifying this:
\[ h + 2k - 3 = -2h - k + 3 \]
\[ 3h + 3k - 6 = 0 \]
\[ h + k = 2 \]

### Step 6: Solve for coordinates using the line equation
We know that the center lies on the line \(3x + 4y = 5\). We can substitute \(k = h\) into this line equation:

For **Case 1**:
\[ 3h + 4h = 5 \Rightarrow 7h = 5 \Rightarrow h = \frac{5}{7}, \quad k = \frac{5}{7} \]
So, one possible coordinate is \(\left(\frac{5}{7}, \frac{5}{7}\right)\).

For **Case 2**:
Substituting \(k = 2 - h\) into the line equation:
\[ 3h + 4(2 - h) = 5 \]
\[ 3h + 8 - 4h = 5 \]
\[ -h + 8 = 5 \Rightarrow -h = -3 \Rightarrow h = 3 \]
Then,
\[ k = 2 - h = 2 - 3 = -1 \]
So, the second possible coordinate is \((3, -1)\).

### Final Answer
The possible coordinates of the centers of the circle are:
1. \(\left(\frac{5}{7}, \frac{5}{7}\right)\)
2. \((3, -1)\)