The sum of abscissa and ordinate of a point on the circle `x^(2)+y^(2)-4x+2y-20=0` which is nearest to `(2, (3)/(2))` is :

Answer

To solve the problem, we need to find the sum of the abscissa (x-coordinate) and ordinate (y-coordinate) of the point on the circle that is nearest to the point (2, 3/2).

### Step 1: Rewrite the equation of the circle
The given equation of the circle is:
\[ x^2 + y^2 - 4x + 2y - 20 = 0 \]

We can rewrite it in standard form by completing the square.

### Step 2: Completing the square for x and y
1. For \(x\):
\[ x^2 - 4x \quad \text{(add and subtract 4)} \]
\[ = (x - 2)^2 - 4 \]

2. For \(y\):
\[ y^2 + 2y \quad \text{(add and subtract 1)} \]
\[ = (y + 1)^2 - 1 \]

Putting it all together:
\[ (x - 2)^2 - 4 + (y + 1)^2 - 1 - 20 = 0 \]
\[ (x - 2)^2 + (y + 1)^2 - 25 = 0 \]
\[ (x - 2)^2 + (y + 1)^2 = 25 \]

### Step 3: Identify the center and radius of the circle
From the standard form \((x - h)^2 + (y - k)^2 = r^2\):
- Center \((h, k) = (2, -1)\)
- Radius \(r = 5\)

### Step 4: Determine the position of the point (2, 3/2)
We need to check if the point (2, 3/2) lies inside or outside the circle. We can do this by calculating the distance from the center to the point and comparing it with the radius.

1. Calculate the distance from the center (2, -1) to the point (2, 3/2):
\[ \text{Distance} = \sqrt{(2 - 2)^2 + \left(\frac{3}{2} - (-1)\right)^2} = \sqrt{0 + \left(\frac{3}{2} + 1\right)^2} = \sqrt{\left(\frac{5}{2}\right)^2} = \frac{5}{2} \]

2. Since \(\frac{5}{2} < 5\), the point (2, 3/2) lies inside the circle.

### Step 5: Find the nearest point on the circle
Since the point (2, 3/2) lies inside the circle, the nearest point on the circle will be directly above or below this point along the vertical line \(x = 2\).

1. The nearest point will be reached by moving a distance equal to the radius (5) from the center (2, -1) in the direction of the y-axis.

2. The center is at (2, -1). Moving 5 units upward:
\[ y = -1 + 5 = 4 \]

Thus, the nearest point on the circle is (2, 4).

### Step 6: Calculate the sum of abscissa and ordinate
The sum of the abscissa and ordinate of the nearest point (2, 4) is:
\[ x + y = 2 + 4 = 6 \]

### Final Answer
The sum of the abscissa and ordinate of the point on the circle which is nearest to (2, 3/2) is **6**.