Question

Find the coordinates of the foot of the perpendicular drawn from the point (-1,3) to the line 3x - 4y -16 =0 .​

Answer 1

Answer:

-31

Step-by-step explanation:

x= -1 y=3

=3(-1) - 4(3) - 16

=-3 -12 -16

= -31

Answer 2

Given:-

• The coordinates of the foot of perpendicular.
• The point (-1, 3) to the line 3x - 4y - 16 = 0

To find:-

• Find the coordinates of the foot of perpendicular.?

Solutions:-

• Let (a, b) be the coordinates of the foot of the perpendicular from the point (-1, 3) to the line 3x - 4y - 16 = 0.

Slope of the line joining (-1, 3) and (a, b),

• m = b - 3 / a + 1

Slope of the line 3x - 4y - 16 = 0 Or y = 3/4 x - 4

• m2 = 3/4

Since the two lines are perpendicular,

• m1m2 = -1

Therefore,

=> (b - 3 / a + 1) × (3/4) = -1

=> 3b - 9 / 4a + 4 = -1

=> 3b - 9 = -4a -

=> 4a + 3b = 5 ...............(i).

Point (a, b) lies on line 3x - 4y = 16

=> 3a - 4b = 16 ..............(ii).

Multiplying Eq. (i) by 3 and Eq. (ii) by 4. We get.

=> 12a + 9b = 15 ........(iii).

=> 12a - 16b = 64 .........(iv).

Subtracting Eq. (iii) from Eq. (iv), we get.

${12a} \: + \: {9b} \: \: = \: \: {15} \\ {12a} \: - \: {16b} \: \: = \: \: {64} \\ \underline{- \: \: \: \: \: \: \: + \: \: \: \: \: \: = \: \: \: \: - \: \: \: \: \: \: \: \: \: \: \: \: \: \: } \\ {25b} \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \: \: \: {-49}$

=> b = -49/25

Putting the value of in Eq. (i).

=> 4a + 3b = 5

=> 4a + 3 × (-49/25) = 5

=> 4a - 147/25 = 5

=> 4a = 5 + 147/25

=> 4a = 125 + 147/25

=> 4a = 272/25

=> a = (272/25)/4

=> a = 68/25

Hence, the required coordinate of the foot of the perpendicular are (68/25, -49/25).