The line perpendicular to the line AB passes through the point 
i.e., 
.
Further explanation:
From the given figure in the question it is observed that the line AB passes through the points 
and 
.
The coordinate for the point C is 
.
Step1: Obtain the slope of the line AB.
The slope of a line which passes through the points 
and 
is calculated as follows:
(1)
It is given that the line AB passes through the points and .
To obtain the slope for the line AB substitute 
for 
, 
for 
, 
for 
and 
for 
in equation (1).
Therefore, the slope of the line AB is 
.
Consider the slope of AB as, 
so, 
.
Step2: Obtain the slope of the perpendicular line.
The slope of line AB is .
Consider a line which is perpendicular to the line AB passing through the point C. Assume the slope of the perpendicular line as 
.
The product of slope of two mutually perpendicular lines is always equal to 
.
The equation formed for the slope is as follows:
Substitute the value of in the above equation.
Therefore, the slope of the perpendicular line is 
.
Step3: Obtain the equation of the perpendicular line.
The slope for perpendicular line is 
and the line passes through the point C. The coordinate for the point C are .
The point slope form of a line is as follows:
Substitute for 
, 
for and for in the above equation.
Therefore, the equation of the perpendicular line is 
.
Option 1:
In option 1 it is given that the line perpendicular to AB passes through the point 
.
The equation of the line which is perpendicular to AB is .
Substitute 
for 
in the above equation.
From the above calculation it is concluded that the line passes through the point 
.
This implies that option 1 is incorrect.
Option 2:
In option 2 it is given that the line perpendicular to AB passes through the point 
.
The equation of the line which is perpendicular to AB is .
Substitute 
for in the above equation.
From the above calculation it is concluded that the line passes through the point .
This implies that option 2 is incorrect.
Option 3:
In option 3 it is given that the line perpendicular to AB passes through the point .
The equation of the line which is perpendicular to AB is .
Substitute for in the above equation.
From the above calculation it is concluded that the line passes through the point .
This implies that option 3 is correct.
Option 4:
In option 4 it is given that the line perpendicular to AB passes through the point 
.
The equation of the line which is perpendicular to AB is .
Substitute for in the above equation.
From the above calculation it is concluded that the line passes through the point 
.
This implies that option 4 is incorrect.
Therefore, the line perpendicular to the line AB passes through the point i.e., 
.
Learn more:
1.A problem on composite function link
2.A problem to find radius and center of circle link
3.A problem to determine intercepts of a line link
Answer details:
Grade: High school
Subject: Mathematics
Chapter: Lines
Keywords: Geometry, coordinate geometry, lines, equation, graph, curve, slope, perpendicular, point slope form, slope intercept form.
