A straight line cuts off the intercepts `OA=a` and `OB=b` on the positive directions of `x` -axis and `y` -axis respectively.If the perpendicular from origin `O` to this line makes an angle of `(pi)/(6)` with positive direction of `y` -axis and the area of `Delta OAB` is `(98)/(3)sqrt(3)`,then `a^(2)-b^(2)` is equal to :

Options

A. `(392)/(3)`
B. `(196)/(3)`
C. `196`
D. `98`

Answer

To solve the given problem step by step, we will follow the information provided in the question and use some geometric properties.

### Step 1: Understand the Geometry
We have a straight line that intercepts the x-axis at point A and the y-axis at point B. The lengths of these intercepts are given as \( OA = a \) and \( OB = b \). The area of triangle OAB is given as \( \frac{98\sqrt{3}}{3} \).

### Step 2: Area of Triangle OAB
The area \( A \) of triangle OAB can be calculated using the formula:
\[ \text{Area} = \frac{1}{2} \times OA \times OB = \frac{1}{2} \times a \times b \]
Setting this equal to the given area:
\[ \frac{1}{2} \times a \times b = \frac{98\sqrt{3}}{3} \]
Multiplying both sides by 2:
\[ a \times b = \frac{196\sqrt{3}}{3} \quad \text{(Equation 1)} \]

### Step 3: Perpendicular from Origin
The perpendicular from the origin O to the line makes an angle of \( \frac{\pi}{6} \) (or 30 degrees) with the positive direction of the y-axis. This means that the angle with the x-axis is \( \frac{\pi}{3} \) (or 60 degrees).

### Step 4: Using Trigonometry
From the right triangle formed by the intercepts and the perpendicular, we can use trigonometric ratios to express the lengths in terms of the angles:
1. For angle \( 60^\circ \):
\[ OP = OA \cos(60^\circ) = a \cdot \frac{1}{2} \quad \text{(Equation 2)} \]
2. For angle \( 30^\circ \):
\[ OP = OB \cos(30^\circ) = b \cdot \frac{\sqrt{3}}{2} \quad \text{(Equation 3)} \]

### Step 5: Equating the Two Expressions for OP
From Equations 2 and 3, we have:
\[ a \cdot \frac{1}{2} = b \cdot \frac{\sqrt{3}}{2} \]
Cancelling \( \frac{1}{2} \) from both sides gives:
\[ a = b\sqrt{3} \quad \text{(Equation 4)} \]

### Step 6: Substitute Equation 4 into Equation 1
Substituting \( a = b\sqrt{3} \) into Equation 1:
\[ (b\sqrt{3}) \cdot b = \frac{196\sqrt{3}}{3} \]
This simplifies to:
\[ b^2 \sqrt{3} = \frac{196\sqrt{3}}{3} \]
Dividing both sides by \( \sqrt{3} \):
\[ b^2 = \frac{196}{3} \]

### Step 7: Find a^2
Now substituting \( b^2 \) back into Equation 4 to find \( a^2 \):
\[ a^2 = 3b^2 = 3 \cdot \frac{196}{3} = 196 \]

### Step 8: Calculate \( a^2 - b^2 \)
Now we can find \( a^2 - b^2 \):
\[ a^2 - b^2 = 196 - \frac{196}{3} \]
Finding a common denominator:
\[ = \frac{588}{3} - \frac{196}{3} = \frac{392}{3} \]

### Final Answer
Thus, the value of \( a^2 - b^2 \) is:
\[ \boxed{\frac{392}{3}} \]