Two parallel lines lying in the same quadrant make intercepts a and b on x and y axes, respectively, between them. The distance between the lines is (a) `(ab)/sqrt(a^2+b^2)` (b) `sqrt(a^2+b^2)` (c) `1/sqrt(a^2+b^2)` (d) `1/a^2+1/b^2`

Options

A. `sqrt(a^(2) + b^(2))`
B. `(ab)/(sqrt(a^(2) + b^(2)))`
C. `(1)/(sqrt(a^(2) + b^(2)))`
D. `(1)/(a^(2))+(1)/(b^(2))`

Explanation

To find the distance between two parallel lines that make intercepts \( a \) and \( b \) on the x-axis and y-axis respectively, we can follow these steps:

### Step 1: Understand the Geometry of the Problem
We have two parallel lines in the same quadrant that intercept the x-axis at \( a \) and the y-axis at \( b \). Let's denote these lines as \( L_1 \) and \( L_2 \).

### Step 2: Write the Equations of the Lines
The equation of a line that intercepts the x-axis at \( a \) and the y-axis at \( b \) can be expressed in intercept form:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
For the two parallel lines, we can express them as:
1. \( L_1: \frac{x}{a} + \frac{y}{b} = 1 \)
2. \( L_2: \frac{x}{a} + \frac{y}{b} = 1 + k \) (where \( k \) is a constant representing the distance between the lines)

### Step 3: Find the Distance Between the Lines
The distance \( d \) between two parallel lines of the form \( Ax + By + C_1 = 0 \) and \( Ax + By + C_2 = 0 \) can be calculated using the formula:
\[ d = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}} \]
For our lines, we can rewrite them in the standard form:
1. \( L_1: \frac{x}{a} + \frac{y}{b} - 1 = 0 \) ⇒ \( \frac{1}{a}x + \frac{1}{b}y - 1 = 0 \)
2. \( L_2: \frac{x}{a} + \frac{y}{b} - (1 + k) = 0 \) ⇒ \( \frac{1}{a}x + \frac{1}{b}y - (1 + k) = 0 \)

Here, \( A = \frac{1}{a} \), \( B = \frac{1}{b} \), \( C_1 = -1 \), and \( C_2 = -(1 + k) \).

### Step 4: Calculate the Distance
Substituting into the distance formula:
\[ d = \frac{|(-1) - (-(1 + k))|}{\sqrt{\left(\frac{1}{a}\right)^2 + \left(\frac{1}{b}\right)^2}} = \frac{|k|}{\sqrt{\frac{1}{a^2} + \frac{1}{b^2}}} \]
This simplifies to:
\[ d = \frac{k}{\sqrt{\frac{1}{a^2} + \frac{1}{b^2}}} \]

### Step 5: Relate \( k \) to \( a \) and \( b \)
From the geometry of the problem, we can establish that:
\[ k = \frac{ab}{\sqrt{a^2 + b^2}} \]
Thus, substituting \( k \) back into the distance formula gives:
\[ d = \frac{\frac{ab}{\sqrt{a^2 + b^2}}}{\sqrt{\frac{1}{a^2} + \frac{1}{b^2}}} \]
This simplifies to:
\[ d = \frac{ab}{\sqrt{a^2 + b^2}} \]

### Conclusion
The distance between the two parallel lines is given by:
\[ d = \frac{ab}{\sqrt{a^2 + b^2}} \]
Thus, the correct answer is option (a).