Let `A=(a,b)` and `B=(c,d)` where `c gt a gt 0` and `d gt b gt 0`. Then, point C on the X-axis such that `AC + BC` is the minimum, is:

Options

A. `(bc-ad)/(b-d)`
B. `(ac+bd)/(b+d)`
C. `(ac-bd)/(b-a)`
D. `(ad+bc)/(b+d)`

Answer

To find the point \( C \) on the X-axis such that the sum of distances \( AC + BC \) is minimized, we can follow these steps:

### Step 1: Understand the Geometry
We have two points \( A = (a, b) \) and \( B = (c, d) \) where \( c > a > 0 \) and \( d > b > 0 \). The point \( C \) lies on the X-axis, which means its coordinates can be represented as \( C = (x, 0) \).

### Step 2: Write the Distance Formulas
The distances from \( A \) to \( C \) and from \( B \) to \( C \) can be expressed as:
- Distance \( AC = \sqrt{(x - a)^2 + (0 - b)^2} = \sqrt{(x - a)^2 + b^2} \)
- Distance \( BC = \sqrt{(x - c)^2 + (0 - d)^2} = \sqrt{(x - c)^2 + d^2} \)

### Step 3: Set Up the Objective Function
We want to minimize the total distance:
\[ D = AC + BC = \sqrt{(x - a)^2 + b^2} + \sqrt{(x - c)^2 + d^2} \]

### Step 4: Use the Reflection Method
To minimize \( D \), we can reflect point \( B \) across the X-axis to get point \( B' = (c, -d) \). The minimum distance \( AC + BC \) is equivalent to the straight line distance \( AB' \).

### Step 5: Find the Equation of the Line
The coordinates of points \( A \) and \( B' \) are \( A = (a, b) \) and \( B' = (c, -d) \). The slope of line \( AB' \) is given by:
\[ \text{slope} = \frac{-d - b}{c - a} \]
The equation of the line can be written in point-slope form:
\[ y - b = \frac{-d - b}{c - a}(x - a) \]

### Step 6: Find the Intersection with the X-axis
To find the point \( C \) on the X-axis, set \( y = 0 \) in the line equation:
\[ 0 - b = \frac{-d - b}{c - a}(x - a) \]
Solving for \( x \):
\[ -b(c - a) = (-d - b)(x - a) \]
\[ -b(c - a) = -dx + da - bx + ba \]
\[ (b + d)x = da + b(c - a) \]
\[ x = \frac{da + b(c - a)}{b + d} \]

### Step 7: Conclusion
Thus, the coordinates of point \( C \) that minimizes \( AC + BC \) are:
\[ C = \left( \frac{da + b(c - a)}{b + d}, 0 \right) \]