Given
The line equation is
![y= \frac{4}{9}x-2](https://tex.z-dn.net/?f=y%3D%20%5Cfrac%7B4%7D%7B9%7Dx-2)
passes through the line (4,3 )
Find out the equation of the perpendicular line.
To proof
given equation is
![y= \frac{4}{9}x-2](https://tex.z-dn.net/?f=y%3D%20%5Cfrac%7B4%7D%7B9%7Dx-2)
the equation of line is in the form y = mx +c
where m = slope
c is the intercept on the y axis.
compare this equation to the above equation
we get
![m =\frac{4}{9}](https://tex.z-dn.net/?f=m%20%3D%5Cfrac%7B4%7D%7B9%7D)
In perpendicular line case
The slope of a perpendicular line is the "negative reciprocal" of the slope of the original line.
thus slope of the perpendicular line
= ![\frac{-9}{4}](https://tex.z-dn.net/?f=%5Cfrac%7B-9%7D%7B4%7D)
Than equation perpendicular line becomes
![y= \frac{-9}{4}x + c](https://tex.z-dn.net/?f=y%3D%20%5Cfrac%7B-9%7D%7B4%7Dx%20%2B%20c)
as the line passes through the point( 4,3)
put these value in the above equation
we get
![3= \frac{-9}{4}\times 4 + c](https://tex.z-dn.net/?f=3%3D%20%5Cfrac%7B-9%7D%7B4%7D%5Ctimes%204%20%2B%20c)
solving the above equation
3 +9 =c
12 =c
put this value in the above equation
we get
![y= \frac{-9}{4}\times x + 12](https://tex.z-dn.net/?f=y%3D%20%5Cfrac%7B-9%7D%7B4%7D%5Ctimes%20x%20%2B%2012)
this is equation of the perpendicular line.
Hence proved