Question

Find the foot of the perpendicular drawn from (4,1)upon the straight line 3x-4y+12​

Answer 1

Solution :

The given straight line is

3x - 4y + 12 = 0 .....(1)

Any line perpendicular to the line (1) can be considered as

4x + 3y = k .....(2) , where k is constant.

If we consider the line (2) to be the perpendicular line to (1) no. line passing through the point (4, 1), then (4, 1) satisfies (2) no. line.

Thus, 4 (4) + 3 (1) = k

or, k = 16 + 3

or, k = 19

Thus, (2) no. line becomes

4x + 3y = 19 .....(3)

After solving, from (1) and (2), we will get the required foot of the perpendicular, as asked.

Two equations are -

3x - 4y = - 12 .....(1)

4x + 3y = 19 .....(3)

Multiplying (1) by 4 and (3) by 3, we get

12x - 16y = - 48

12x + 9y = 57

On subtraction, we get

- 16y - 9y = - 48 - 57

or, 25y = 105

or, y = 105/25

or, y = 21/5

Also, x = (19 - 3y)/4, by (3)

= {19 - 3 (21/5)}/4

= {19 - 63/5}/4

= {(95 - 63)/5}/4

= (32/5)/4

= 8/5

or, x = 8/5

∴ the foot of the perpendicular is (8/5, 21/5).