Two towns A and B are situated on the same side of a straight road at distances a and brespectively perpendiculars drawn from A and B meet the road at point C and D respectively.The distance between C and D is c. A hospital is to be built at a point P on the road such thatthe distance APB is minimum. Find position of P.

Answer

To solve the problem of finding the optimal position of point P on the road such that the distance APB is minimized, we can follow these steps:

### Step 1: Understand the Geometry
We have two points A and B located at distances a and b from the road, respectively. The perpendiculars from A and B to the road meet the road at points C and D, respectively. The distance between points C and D is c.

### Step 2: Define the Position of Point P
Let the position of point P on the road be at a distance x from point C. Therefore, the distance from P to C is x and the distance from P to D is (c - x).

### Step 3: Express Distances AP and PB
Using the Pythagorean theorem, we can express the distances AP and PB as follows:
- Distance AP = √(a² + x²)
- Distance PB = √(b² + (c - x)²)

### Step 4: Write the Total Distance APB
The total distance APB can be expressed as:
\[ D(x) = AP + PB = \sqrt{a^2 + x^2} + \sqrt{b^2 + (c - x)^2} \]

### Step 5: Differentiate D(x)
To find the minimum distance, we need to differentiate D(x) with respect to x:
\[ D'(x) = \frac{x}{\sqrt{a^2 + x^2}} - \frac{(c - x)}{\sqrt{b^2 + (c - x)^2}} \]

### Step 6: Set D'(x) to Zero
To find the critical points, we set the derivative equal to zero:
\[ \frac{x}{\sqrt{a^2 + x^2}} = \frac{(c - x)}{\sqrt{b^2 + (c - x)^2}} \]

### Step 7: Solve for x
Cross-multiplying gives:
\[ x \sqrt{b^2 + (c - x)^2} = (c - x) \sqrt{a^2 + x^2} \]
Squaring both sides and simplifying will yield a quadratic equation in terms of x.

### Step 8: Find the Value of x
After solving the quadratic equation, we can find the value of x that minimizes the distance D(x).

### Step 9: Determine the Position of P
The position of P on the road can then be determined as:
\[ P = C + x \]

### Final Answer
Thus, the position of point P on the road that minimizes the distance APB can be calculated using the derived value of x.

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