Question

Equation of a line passing through \[(1, - 2)\] and perpendicular to the line \[3x - 5y + 7 = 0\] is
A. \[5x + 3y + 1 = 0\]
В. \[3x + 5y + 1 = 0\]
C. \[5x - 3y - 1 = 0\]
D. \[3x - 5y + 1 = 0\]

Answer 1

Hint: To get the equation of a line, use the knowledge that the sum of the slopes of any two perpendicular lines equals \[ - 1\] and to construct the equation of a line passing through a given point with a particular slope, use the one point form of the line.

Formula Used: The slope of the line in the form of \[ax + by = c\] is
\[\dfrac{{ - a}}{b}\]

Complete step by step solution: We have been provided in the question that,
It passes through a point
\[(1, - 2)\]
And it is perpendicular to the line
\[3x - 5y + 7 = 0\]
Now, we have to determine the equation perpendicular to \[3x - 5y + 7 = 0\]
The equation that is perpendicular to \[3x - 5y + 7 = 0\]is
\[5x + 3y + \lambda = 0\]---- (1)
We already know that the equation passes through a point \[(1, - 2)\]
No, we have to substitute the points in the equation (1),
Now, it becomes,
\[5 \times 1 + 3 \times ( - 2) + \lambda = 0\]
On simplifying the above equation, we get
\[5 - 6 + \lambda = 0\]
Now, we have to solve for \[\lambda \] by having it on one side and solving the remaining terms, we have
\[\lambda = 1\]
Now, we have to replace the value of \[\lambda \] on to the equation (1) to obtain the desired equation, we get
\[5x + 3y + 1 = 0\]
Therefore, equation of a line passing through \[(1, - 2)\] and perpendicular to the line \[3x - 5y + 7 = 0\] is \[5x + 3y + 1 = 0\]

Option ‘A’ is correct

Note: The equation of a line must be written in point slope form using the knowledge that the sum of the slopes of any two perpendicular lines is \[ - 1\] and without employing these data, we won't be able to write the equation for a line. Any line equation describes the connection between two points on a line. The point slope form of the line connects any basic point on the line with the slope of the line