Circle through the point M (5, 4) and touching x-axis at L (2,0) is ..........

Answer

To find the equation of the circle that passes through the point M(5, 4) and touches the x-axis at the point L(2, 0), we can follow these steps:

### Step 1: Determine the center of the circle
Since the circle touches the x-axis at L(2, 0), the center of the circle must be directly above this point on the y-axis. Let the center of the circle be (h, k). Since the circle touches the x-axis, the y-coordinate of the center (k) is equal to the radius (r) of the circle.

From the point L(2, 0), we have:
- h = 2 (the x-coordinate of the point where the circle touches the x-axis)
- k = r (the radius of the circle)

Thus, the center of the circle can be represented as (2, r).

### Step 2: Use the point M(5, 4) to find the radius
The circle passes through the point M(5, 4). Therefore, the distance from the center (2, r) to the point M(5, 4) must equal the radius r. We can use the distance formula to set up the equation:

\[ \sqrt{(5 - 2)^2 + (4 - r)^2} = r \]

### Step 3: Simplify the equation
Squaring both sides to eliminate the square root gives us:

\[ (5 - 2)^2 + (4 - r)^2 = r^2 \]

Calculating (5 - 2)^2:

\[ 3^2 + (4 - r)^2 = r^2 \]

This simplifies to:

\[ 9 + (4 - r)^2 = r^2 \]

### Step 4: Expand and rearrange the equation
Now, expand (4 - r)^2:

\[ 9 + (16 - 8r + r^2) = r^2 \]

Combining like terms gives:

\[ 25 - 8r = 0 \]

### Step 5: Solve for r
Now, solve for r:

\[ 8r = 25 \implies r = \frac{25}{8} \]

### Step 6: Determine the center coordinates
Now that we have r, we can find the center of the circle:

\[ h = 2, \quad k = r = \frac{25}{8} \]

So the center of the circle is (2, \(\frac{25}{8}\)).

### Step 7: Write the equation of the circle
The standard form of the equation of a circle is:

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

Substituting the values of h, k, and r:

\[ (x - 2)^2 + \left(y - \frac{25}{8}\right)^2 = \left(\frac{25}{8}\right)^2 \]

Calculating \(\left(\frac{25}{8}\right)^2\):

\[ \left(\frac{25}{8}\right)^2 = \frac{625}{64} \]

Thus, the equation of the circle is:

\[ (x - 2)^2 + \left(y - \frac{25}{8}\right)^2 = \frac{625}{64} \]

### Final Answer
The equation of the circle is:

\[ (x - 2)^2 + \left(y - \frac{25}{8}\right)^2 = \frac{625}{64} \]

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