The equation of the circle which passes through the points `(2,3) and (4,5) ` and the centre lies on the straight line `y-4x + 3=0,` is

Options

A. `x ^(2) + y^(2) + 4x -10y + 25 =0`
B. `x ^(2) + y^(2) -4x-10y + 25 =0`
C. `x ^(2) + y^(2) -4x -10 y + 16=0`
D. `x ^(2) + y^(2) - 14 y + 8=0`

Answer

To find the equation of the circle that passes through the points (2, 3) and (4, 5) with its center lying on the line \( y - 4x + 3 = 0 \), we can follow these steps:

### Step 1: Define the center of the circle
Let the center of the circle be \( (h, k) \). Since the center lies on the line \( y - 4x + 3 = 0 \), we can express this as:
\[ k = 4h - 3 \]

### Step 2: Use the distance formula
The distance from the center \( (h, k) \) to the point \( (2, 3) \) must be equal to the distance from the center to the point \( (4, 5) \). Using the distance formula, we have:
\[ \sqrt{(h - 2)^2 + (k - 3)^2} = \sqrt{(h - 4)^2 + (k - 5)^2} \]

### Step 3: Square both sides
Squaring both sides to eliminate the square root gives us:
\[ (h - 2)^2 + (k - 3)^2 = (h - 4)^2 + (k - 5)^2 \]

### Step 4: Expand both sides
Expanding both sides:
\[ (h^2 - 4h + 4) + (k^2 - 6k + 9) = (h^2 - 8h + 16) + (k^2 - 10k + 25) \]
This simplifies to:
\[ h^2 - 4h + 4 + k^2 - 6k + 9 = h^2 - 8h + 16 + k^2 - 10k + 25 \]

### Step 5: Cancel common terms
Cancelling \( h^2 \) and \( k^2 \) from both sides:
\[ -4h + 4 - 6k + 9 = -8h + 16 - 10k + 25 \]
This simplifies to:
\[ -4h - 6k + 13 = -8h - 10k + 41 \]

### Step 6: Rearrange the equation
Rearranging gives:
\[ 4h + 4k - 28 = 0 \]
Dividing by 4:
\[ h + k - 7 = 0 \quad \text{(Equation 1)} \]

### Step 7: Substitute \( k \) from the line equation
Substituting \( k = 4h - 3 \) into Equation 1:
\[ h + (4h - 3) - 7 = 0 \]
This simplifies to:
\[ 5h - 10 = 0 \]
Thus, we find:
\[ h = 2 \]

### Step 8: Find \( k \)
Substituting \( h = 2 \) back into the equation for \( k \):
\[ k = 4(2) - 3 = 8 - 3 = 5 \]

### Step 9: Find the radius
The center of the circle is \( (2, 5) \). Now, we calculate the radius using the distance from the center to one of the points, say \( (2, 3) \):
\[ r = \sqrt{(2 - 2)^2 + (5 - 3)^2} = \sqrt{0 + 4} = 2 \]

### Step 10: Write the equation of the circle
The general equation of a circle is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \( h = 2 \), \( k = 5 \), and \( r = 2 \):
\[ (x - 2)^2 + (y - 5)^2 = 2^2 \]
This simplifies to:
\[ (x - 2)^2 + (y - 5)^2 = 4 \]

### Step 11: Expand the equation
Expanding this gives:
\[ x^2 - 4x + 4 + y^2 - 10y + 25 = 4 \]
Simplifying further:
\[ x^2 + y^2 - 4x - 10y + 25 = 0 \]

Thus, the equation of the circle is:
\[ x^2 + y^2 - 4x - 10y + 25 = 0 \]