The sum of two non-negative numbers is 2a. If P be the probability that the product of these numbers is not less than m times their greatest possible product, then

Options

A. `P=(2)/(3),m=(4)/(5)`
B. `P=(1)/(3), m=(3)/(5)`
C. `P=(1)/(2),m=(3)/(4)`
D. `P=(3)/(4),m=(7)/(16)`.

Answer

To solve the problem step by step, we will analyze the conditions given and derive the probability \( P \) that the product of two non-negative numbers is not less than \( m \) times their greatest possible product.

### Step 1: Define the variables
Let the two non-negative numbers be \( x \) and \( 2a - x \). The sum of these two numbers is given as \( 2a \).

### Step 2: Find the greatest possible product
The product \( P(x) \) of these two numbers is:
\[ P(x) = x(2a - x) = 2ax - x^2 \]
To find the maximum product, we can set the derivative of \( P(x) \) to zero:
\[ \frac{dP}{dx} = 2a - 2x = 0 \implies x = a \]
Thus, the greatest possible product occurs when both numbers are equal, which gives us:
\[ P(a) = a(2a - a) = a^2 \]

### Step 3: Set up the inequality
We need to find the probability that the product \( P(x) \) is not less than \( m \) times the greatest possible product:
\[ x(2a - x) \geq m a^2 \]
This simplifies to:
\[ 2ax - x^2 \geq ma^2 \]
Rearranging gives:
\[ x^2 - 2ax + ma^2 \leq 0 \]

### Step 4: Solve the quadratic inequality
This is a quadratic inequality in \( x \). The roots of the corresponding equation \( x^2 - 2ax + ma^2 = 0 \) can be found using the quadratic formula:
\[ x = \frac{2a \pm \sqrt{(2a)^2 - 4 \cdot 1 \cdot ma^2}}{2 \cdot 1} = a \pm a\sqrt{1 - m} \]
Thus, the roots are:
\[ x_1 = a(1 - \sqrt{1 - m}), \quad x_2 = a(1 + \sqrt{1 - m}) \]

### Step 5: Determine the range of \( x \)
The quadratic \( x^2 - 2ax + ma^2 \leq 0 \) holds between the roots:
\[ a(1 - \sqrt{1 - m}) \leq x \leq a(1 + \sqrt{1 - m}) \]

### Step 6: Calculate the favorable cases
The length of the interval where the product is not less than \( m a^2 \) is:
\[ \text{Length} = x_2 - x_1 = a(1 + \sqrt{1 - m}) - a(1 - \sqrt{1 - m}) = 2a\sqrt{1 - m} \]

### Step 7: Calculate the total cases
The total possible values for \( x \) range from \( 0 \) to \( 2a \), giving a total length of \( 2a \).

### Step 8: Find the probability \( P \)
The probability \( P \) that the product is not less than \( m \) times the greatest possible product is given by the ratio of favorable cases to total cases:
\[ P = \frac{\text{Favorable cases}}{\text{Total cases}} = \frac{2a\sqrt{1 - m}}{2a} = \sqrt{1 - m} \]

### Final Result
Thus, the probability \( P \) is:
\[ P = \sqrt{1 - m} \]