Question

The equation of the line perpendicular to 2x – 3y + 5 = 0 and making an intercept 3 with positive Y-axis is

• A. 3x + 2y – 6 = 0
• B. 3x + 2y – 12 = 0
• C. 3x + 2y – 7 = 0
• D. 3x + 2y + 6 = 0

Answer 1

The Correct Option is A

Concept:
To find the equation of a line that is perpendicular to a given line and also passes through a specific point (or intercept), we need to use the relationship between slopes and apply the general equation of a straight line: y = mx + c.

Step 1: Find the slope of the given line
The given line is: 2x − 3y + 5 = 0
To find the slope, we rearrange this in the slope-intercept form: y = mx + c
2x − 3y + 5 = 0 ⟹ −3y = −2x − 5 ⟹ y = (2/3)x + 5/3
So, the slope of the given line is 2/3.

Step 2: Find the slope of the perpendicular line
If two lines are perpendicular, then the product of their slopes is −1.
So, the slope of the required perpendicular line is:
m = −1 / (2/3) = −3/2

Step 3: Use the given condition (y-intercept = 3)
We are told that the required line cuts the y-axis at 3, i.e., c = 3.

Now, using the slope-intercept form:
y = mx + c
⇒ y = −(3/2)x + 3

Step 4: Convert the equation into standard form
Multiply the entire equation by 2 to eliminate the denominator:
2y = −3x + 6
Rearranging: 3x + 2y − 6 = 0

Conclusion:
The equation of the line that is perpendicular to the given line and has a y-intercept of 3 is:
3x + 2y − 6 = 0

Therefore, the correct option is: (A) 3x + 2y − 6 = 0.