Question

A line passes through the point (3, 1) and cuts off positive intercepts on the X-axis and Y-axis in the ratio 2 : 3. Find the equation of the line.

Answer 1

Let the x-intercept be a and the y-intercept be b.

Since the ratio of the intercepts is 2 : 3,

We can represent them as:

⇒ x-intercept (a): 2k

⇒ y-intercept (b): 3k

Using the intercept formula:

`x/a + y/b = 1`

Substituting a = 2k and b = 3k:

`x/(2k) + y/(3k) = 1`

The line passes through the point (3, 1),

Substitute x = 3 and y = 1 in the equation,

`3/(2k) + 1/(3k) = 1`

`9/(6k) + 2/(6k) = 1`   ...[Cross-multiplied to find the common denominator.]

`11/(6k) = 1`

6k = 11

∴ k = `11/6`

Substitute k = `11/6` back into the intercept values:

`a = 2 xx 11/6`

∴ a = `11/3`

`b = 3 xx 11/6`

∴ b = `11/2`

Now, put these into the intercept form:

`x/(11/3) + y/(11/2) = 1`

`(3x)/11 + (2y)/11 = 1`

3x + 2y = 11   ...[Multiplied the entire equation by 11]

Let’s write the above equation in standard form (Ax + By + C = 0),

3x + 2y − 11 = 0

Hence, the equation of the line is 3x + 2y − 11 = 0 or 3x + 2y = 11.