A line \[AB\] makes zero intercepts on x-axis and y-axis and it is perpendicular to another line \[CD\]
And \[3x + 4y + 6 = 0\] then the equation of line \[AB\] is
A. \[y = 4\]
B. \[4x - 3y + 8 = 0\]
C. \[4x - 3y = 0\]
D. \[4x - 3y + 6 = 0\]

Hint: Graphs can help us tackle this challenge. We can also tackle this problem by reducing options utilizing specific strategies. We can get the proper option in either case. However, determining the slope and a point passing through it is how to answer this question.

Formula Used: The slope of a line with the equation \[ax + by + c = 0\] is,
\[ \Rightarrow \dfrac{{ - a}}{b}\]

Complete step by step solution: We have been given in the question that a line \[AB\] makes zero intercepts on x-axis and y-axis and it is perpendicular to another line \[CD\]
From the given statement, it is understood that, it passes through the origin
\[(0,0)\]
We must now obtain the slope in order to compute the equation of the needed line.
And the equation provided is,
\[3x + 4y + 6 = 0\]
Consequently, the equation \[3x + 4y + 6 = 0\] slope will be,
\[ \Rightarrow - \dfrac{3}{4}\]
From this, the slope of AB will be determined as,
\[ \Rightarrow \dfrac{4}{3}\]
As we already know that the equation for straight line is,
\[y = mx + c\]
As a result, the necessary equation crossing through the origin and slope m is,
\[y = \dfrac{4}{3}x\]
Now, let’s convert the improper fraction by multiplying the denominator to the other side of the equation, we get
\[3y - 4x = 0\]
The preceding calculation can alternatively be expressed as,
\[4x - 3y = 0\]
Therefore, the equation of line \[AB\] is \[4x - 3y = 0\].

Option ‘C’ is correct

Note: This is a problem that can help you save time during the exam. We can find the solution by considering the alternatives and eliminating them. We've already established that the needed line has zero intercepts on both axes. In the above possibilities, there is only one option with zero intercepts on both axes. We can save time in the examination by employing this method.