A circle passes through the points ( 2,3) and ( 4,5) . If its centre lies on the line, ` y-4x+ 3 =0`, then its radius is equal to `:`

Options

A. 1
B. 2
C. `sqrt(5)`
D. `sqrt(2)`

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step 1: Set up the general equation of the circle
The general equation of a circle can be written as:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
where the center of the circle is at the point \((-g, -f)\) and the radius \(R\) is given by:
\[ R = \sqrt{g^2 + f^2 - c} \]

### Step 2: Use the points the circle passes through
The circle passes through the points \((2, 3)\) and \((4, 5)\). We will substitute these points into the general equation to form two equations.

#### For the point (2, 3):
Substituting \(x = 2\) and \(y = 3\):
\[ 2^2 + 3^2 + 2g(2) + 2f(3) + c = 0 \]
This simplifies to:
\[ 4 + 9 + 4g + 6f + c = 0 \implies 13 + 4g + 6f + c = 0 \tag{1} \]

#### For the point (4, 5):
Substituting \(x = 4\) and \(y = 5\):
\[ 4^2 + 5^2 + 2g(4) + 2f(5) + c = 0 \]
This simplifies to:
\[ 16 + 25 + 8g + 10f + c = 0 \implies 41 + 8g + 10f + c = 0 \tag{2} \]

### Step 3: Eliminate \(c\)
To eliminate \(c\), we subtract equation (1) from equation (2):
\[ (41 + 8g + 10f + c) - (13 + 4g + 6f + c) = 0 \]
This simplifies to:
\[ 28 + 4g + 4f = 0 \implies 4g + 4f = -28 \implies g + f = -7 \tag{3} \]

### Step 4: Use the line condition
The center of the circle \((-g, -f)\) lies on the line \(y - 4x + 3 = 0\). Substituting \(-g\) for \(x\) and \(-f\) for \(y\):
\[ -f - 4(-g) + 3 = 0 \implies -f + 4g + 3 = 0 \implies 4g - f = -3 \tag{4} \]

### Step 5: Solve the system of equations
Now we have two equations (3) and (4):
1. \(g + f = -7\)
2. \(4g - f = -3\)

We can add these two equations to eliminate \(f\):
\[ (g + f) + (4g - f) = -7 - 3 \]
This simplifies to:
\[ 5g = -10 \implies g = -2 \]

Now substituting \(g = -2\) back into equation (3):
\[ -2 + f = -7 \implies f = -5 \]

### Step 6: Find \(c\)
Now we substitute \(g\) and \(f\) back into equation (1) to find \(c\):
\[ 13 + 4(-2) + 6(-5) + c = 0 \]
This simplifies to:
\[ 13 - 8 - 30 + c = 0 \implies c = 25 \]

### Step 7: Calculate the radius
Now we can find the radius using the formula:
\[ R = \sqrt{g^2 + f^2 - c} \]
Substituting the values of \(g\), \(f\), and \(c\):
\[ R = \sqrt{(-2)^2 + (-5)^2 - 25} = \sqrt{4 + 25 - 25} = \sqrt{4} = 2 \]

### Final Answer
The radius of the circle is \(2\).

---