Find the equation of a line passes through `(3,4)` and the ratio of its intercepts on `X` and `Y`-axis is `3 : 2`.

Answer

To find the equation of a line that passes through the point (3, 4) and has intercepts on the x-axis and y-axis in the ratio of 3:2, we can follow these steps:

### Step 1: Understand the intercept form of the line
The equation of a line in intercept form is given by:
\[ \frac{x}{a} + \frac{y}{b} = 1 \]
where \(a\) is the x-intercept and \(b\) is the y-intercept.

### Step 2: Define the intercepts based on the ratio
Given that the ratio of the intercepts is \(3:2\), we can express the intercepts as:
- Let the x-intercept \(a = 3k\)
- Let the y-intercept \(b = 2k\)

for some constant \(k\).

### Step 3: Substitute the intercepts into the intercept form
Substituting \(a\) and \(b\) into the intercept form, we have:
\[ \frac{x}{3k} + \frac{y}{2k} = 1 \]

### Step 4: Clear the denominators
To eliminate the denominators, multiply the entire equation by \(6k\):
\[ 6k \left(\frac{x}{3k}\right) + 6k \left(\frac{y}{2k}\right) = 6k \]
This simplifies to:
\[ 2x + 3y = 6k \]

### Step 5: Use the point (3, 4) to find \(k\)
Since the line passes through the point (3, 4), we can substitute \(x = 3\) and \(y = 4\) into the equation:
\[ 2(3) + 3(4) = 6k \]
Calculating the left side:
\[ 6 + 12 = 6k \]
So,
\[ 18 = 6k \]
This gives us:
\[ k = 3 \]

### Step 6: Substitute \(k\) back into the equation
Now substitute \(k = 3\) back into the equation \(2x + 3y = 6k\):
\[ 2x + 3y = 6(3) \]
This simplifies to:
\[ 2x + 3y = 18 \]

### Final Answer
The equation of the line is:
\[ \boxed{2x + 3y = 18} \]

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