Correct Answer - Option 4 :
\(\left( {\frac{{78}}{{41}},\frac{{128}}{{41}}} \right)\)
Calculation:
Let (h, k) be the coordinates of the foot of the perpendicular from the point (2, 3) on the line 4x - 5y + 8 = 0.
The slope of the perpendicular line will be \(\frac{k - 3}{h - 2}\)
The slope of the line 4x - 5y + 8 = 0
⇒ 4x - 5y + 8 = 0
⇒ -5y = -4x - 8
⇒ y = 4/5x + 8/5
On comparing the above equation with y = mx + c we get m = 4/5
∴ the slope of the line is 4/5
Using the condition of perpendicularity of lines,
\(\frac{k - 3}{h - 2} \times \frac{4}{5} = -1\)
4k - 12 = - 5h + 10
4k + 5h = 22 ----(i)
Since (h, k) lies on the given line 4x - 5y + 8 = 0,
⇒ 4h - 5k + 8 = 0
⇒ 4h - 5k = - 8 ----(ii)
on solving the equations (i) and (ii) using elimination method, we get h = 78/41 and k = 128/41
Therefore, \(\left( {\frac{{78}}{{41}},\frac{{128}}{{41}}} \right)\)are the coordinates of the foot of the perpendicular from the point (2, 3) on the line 4x - 5y +8 = 0.
