Question

A line passes through the point \((3,4)\) and cuts off intercepts from the coordinates axes such that their sum is \[14\]. The equation of the line is

A) \[4x - 3y = 24\]
B) \[\;\;4x + 3y = 24\]
C) \[\;3x - 4y = 24\]
D) \[\;3x + 4y = 24\]

Answer 1

Hint: Straight line is a set of infinites points in which all points are linear. Intercept is a point where a line cuts the x or y axis. In this question we have to find the equation of line which intercept equally on both axes. As intercept is given in this question therefore equation of the intercept form of straight line will be used in this question. Points lying on particular lines must satisfy the equation of the line.



Formula Used:In this question equation of intercept form of straight line is used:
\(\dfrac{x}{a} + \dfrac{y}{b} = 1\)
Where,
a and b are intercept of x and y axis respectively



Complete step by step solution:Given : straight line passes through the point \((3,4)\) and intercept are equal.

The equation of intercept form of straight line is
\(\dfrac{x}{a} + \dfrac{y}{b} = 1\)
a and b are intercept of x and y axis respectively
According to question intercept are equal
\[a + b = 14\]
Equation of required line
\(\dfrac{x}{{14 - b}} + \dfrac{y}{b} = 1\)
Now this lines passes through \((3,4)\)so this coordinate must satisfy the equation of line
\(\dfrac{3}{{14 - b}} + \dfrac{4}{b} = 1\)
\(3b + 56 - 4b = (14 - b)b\)
On simplification
\(b = 7\)Or \(b = 8\)
Equation of required line is:
When \(b = 7\)
\(\dfrac{x}{7} + \dfrac{y}{7} = 1\)
\(x + y = 7\)
When \(b = 8\)
\(\dfrac{x}{6} + \dfrac{y}{8} = 1\)
\[4x + 3y = 24\]



Option ‘B’ is correct



Note: Do not use the equation of line in any other form because it will become very difficult to find the equation of lines and sometimes one may not find the equation of line by using the general equation of lines. If in any question an intercept on line is given then use only the intercept form of the straight line equation.