The line `2y = ax+b` is a tangent to parabola `y^(2) = 4ax(a in N)`, then for maximum value of` [a+b/a]` the value of 'a' is ({.] denotes the greates integer function)

Options

A. 2
B. 3
C. 1
D. 4

Answer

To solve the problem step by step, we need to establish the relationship between the line and the parabola, and then maximize the expression \(\left\lfloor \frac{a + b}{a} \right\rfloor\).

### Step 1: Write the equations
The equation of the parabola is given by:
\[ y^2 = 4ax \]
The equation of the line is given by:
\[ 2y = ax + b \quad \Rightarrow \quad y = \frac{ax + b}{2} \]

### Step 2: Find the point of tangency
For the line to be a tangent to the parabola, it must touch the parabola at exactly one point. We can substitute \(y\) from the line equation into the parabola equation.

Substituting \(y\) in the parabola's equation:
\[ \left(\frac{ax + b}{2}\right)^2 = 4ax \]
Expanding this gives:
\[ \frac{(ax + b)^2}{4} = 4ax \]
Multiplying through by 4 to eliminate the fraction:
\[ (ax + b)^2 = 16ax \]
Expanding the left side:
\[ a^2x^2 + 2abx + b^2 = 16ax \]
Rearranging gives us a quadratic equation in \(x\):
\[ a^2x^2 + (2ab - 16a)x + b^2 = 0 \]

### Step 3: Condition for tangency
For the line to be tangent to the parabola, the discriminant of this quadratic must be zero:
\[ (2ab - 16a)^2 - 4a^2b^2 = 0 \]
Simplifying this:
\[ (2ab - 16a)^2 = 4a^2b^2 \]
Taking the square root of both sides:
\[ |2ab - 16a| = 2ab \]
This leads to two cases:
1. \(2ab - 16a = 2ab\) which simplifies to \(16a = 0\) (not possible since \(a\) is a natural number).
2. \(2ab - 16a = -2ab\) which simplifies to \(4ab = 16a\) or \(b = 4\) (assuming \(a \neq 0\)).

### Step 4: Substitute \(b\) back
Now we have \(b = 4\). We need to find the maximum value of:
\[ \frac{a + b}{a} = \frac{a + 4}{a} = 1 + \frac{4}{a} \]
To maximize this expression, we need to minimize \(a\) since \(a\) is a natural number.

### Step 5: Find maximum value
The expression \(1 + \frac{4}{a}\) increases as \(a\) decreases. The smallest natural number is \(1\):
\[ \text{If } a = 1, \quad 1 + \frac{4}{1} = 5 \]
\[ \text{If } a = 2, \quad 1 + \frac{4}{2} = 3 \]
\[ \text{If } a = 3, \quad 1 + \frac{4}{3} \approx 2.33 \]
\[ \text{If } a = 4, \quad 1 + \frac{4}{4} = 2 \]
As \(a\) increases, the value of the expression decreases.

### Step 6: Greatest Integer Function
Thus, the maximum value of \(\frac{a + b}{a}\) occurs at \(a = 1\) and is equal to \(5\). Therefore, the greatest integer function value is:
\[ \left\lfloor 5 \right\rfloor = 5 \]

### Final Answer
The value of \(a\) for which \(\left\lfloor \frac{a + b}{a} \right\rfloor\) is maximized is:
\[ \boxed{1} \]