Equation of a circle which touches y-axis at (0,2) and cuts ff an intercept of 3 units from the x-axis is `x ^(2) + y ^(2) - 2 alpha x -4y + 4=0` where `alpha ^(2)=`

Options

A. `5//2`
B. `25//4`
C. `4//25`
D. `29`

Answer

To find the value of \( \alpha^2 \) for the given circle equation, we will follow these steps:

### Step 1: Understand the properties of the circle
The circle touches the y-axis at the point (0, 2). This means that the distance from the center of the circle to the y-axis is equal to the radius of the circle.

### Step 2: Define the center and radius
Let the center of the circle be \( (h, k) \). Since the circle touches the y-axis at (0, 2), we know:
- The y-coordinate of the center \( k = 2 \).
- The radius \( r \) is equal to the x-coordinate of the center \( |h| \).

### Step 3: Determine the x-intercept
The circle cuts off an intercept of 3 units from the x-axis. This means the distance from the center to the x-axis is \( k = 2 \), and the total distance from the x-axis to the points where the circle intersects the x-axis is 3 units. Therefore, the distance from the center to the x-axis must be equal to half of the intercept, which is \( \frac{3}{2} \).

### Step 4: Set up the relationship
Since the radius \( r \) is equal to \( |h| \) and the distance from the center to the x-axis is \( k = 2 \), we can write:
\[ r = |h| = 2 + \frac{3}{2} = \frac{7}{2} \]

### Step 5: Calculate \( h \)
From the above equation, we have:
\[ |h| = \frac{7}{2} \]
Thus, \( h = \frac{7}{2} \) or \( h = -\frac{7}{2} \).

### Step 6: Write the equation of the circle
The general equation of a circle is given by:
\[ (x - h)^2 + (y - k)^2 = r^2 \]
Substituting \( k = 2 \) and \( r = \frac{7}{2} \):
\[ (x - h)^2 + (y - 2)^2 = \left(\frac{7}{2}\right)^2 \]

### Step 7: Expand the equation
Expanding the equation:
\[ (x - h)^2 + (y - 2)^2 = \frac{49}{4} \]
This expands to:
\[ x^2 - 2hx + h^2 + y^2 - 4y + 4 = \frac{49}{4} \]
Rearranging gives:
\[ x^2 + y^2 - 2hx - 4y + \left(h^2 + 4 - \frac{49}{4}\right) = 0 \]

### Step 8: Compare with the given equation
The given equation is:
\[ x^2 + y^2 - 2\alpha x - 4y + 4 = 0 \]
From this, we can identify:
- \( 2\alpha = 2h \) ⇒ \( \alpha = h \)
- The constant term gives us \( h^2 + 4 - \frac{49}{4} = 4 \)

### Step 9: Solve for \( h^2 \)
From the constant term:
\[ h^2 + 4 - \frac{49}{4} = 4 \]
This simplifies to:
\[ h^2 - \frac{49}{4} = 0 \]
Thus:
\[ h^2 = \frac{49}{4} \]

### Step 10: Find \( \alpha^2 \)
Since \( \alpha = h \), we have:
\[ \alpha^2 = h^2 = \frac{49}{4} \]

### Final Answer
Thus, the value of \( \alpha^2 \) is:
\[ \alpha^2 = \frac{49}{4} \]