# Find the equation of the perpendicular from P(-1,-2) on the line 3x+4y=12. Also find the coordinates of the foot of perpendicular.

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3x+4y is the equation of a straight line whose slope equals -a/b where a is the coefficient of x and b is the coefficient of y which is equal to (-3/4)

since the line which is required to find its equation is perpendicular to that line , therefore its slope equals ( 4/3 )[ switch and invert ]

to find the equation of that line u have now its slope which is ( 4/3 )and a point on it (-1,-2) so use the slope-point form y-(-2)=4/3(x-(-1))

y+2=4/3 (x+1) *3

3y+6 = 4 (x+1)

3y+6 = 4x+4

4x-3y = 6 - 4

4x+3y = 2

to find the foot of perpendicular solve the equations of the two line together to get their point of intersection

4x + 3y = 2 ( times -4 ) therefore -16x - 12y = -8

3x + 4y =12 ( times 3 ) therefore 9x + 12y = 36

by adding the two equations therefore

- 7x = 28 which leads to x = -4

if we substitute in one of the two equations

therefore 4 ( -4 ) + 3y = 2 leads to -16 + 3y = 2

and hence 3y = 2 + 16 --------------- 3y = 18

and therefore y = 6 the foot will be ( -4 , 6 )

If we represent the equation of a straight line in the form y = mx = c,

m represents the slope of the equation.

The slope of the line perpendicular to this line is -1/m.

converting the equation

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

to the form y = mx + c we get:

y = (-3/4)x + 3

Therefore slope of a line perpendicular to it will be 4/3.

and general equation of all the line perpendicular to the original line is given by the equation:

y = (4/3)x + c

Given that the perpendicular passes through the point (-1, -2), we substitute these values of coordinates in the above equation. Thus:

-2 = (4/3)*(-1) + c

Or

c = -2 + 4/3 = -2/3

Substituting this value of c in general equation of perpendicular we get:

y = 4x/3 - 2/3

this is same as:

4x - 3y = 2 ... (2)

To find the point of intersection we simultaneously solve equation (1) and (2).

Multiplying equation (1) by 4 and equation (2) by 3 we get:

12x + 16y = 48 ... (3)

12x - 9y = 6 ... (4)

Subtracting equation (4) from (5)

25y = 42

therefore y = 42/25

substituting this value of y in equation (2):

4x - 3*(42/25) = 2

Therefore

x = (2 + 126/25)/4 = 44/25

Therefore the line and its perpendicular intersect at the point (44/25, 42/25)

Any line passing through (x1,y1) is y=m(x-x1), where m is the slope of the line.Therefore, the line through P(-1,-2) is

y = m(x+1)-2 . .........(1). The given line 3x+4y=12 in slope intercept form is y = -(3/4)x-2......(2).Since the line (1) is perpendicular to (2), it requires that the product of their slope is -1 Or

m(-3/4) =-1 . Or m = (4/3). Substuting the value of m = 4/3 in equation (1), we get: y = (4/3)(x+1)-2 = (4/3)x-2/3 . Or

y=(4/3)x-2/3 is the required perpendicular line in slope intercept form. Or 4x-3y - 2 = 0.........(3) in the standard ax+by+c =0 form.

The feet of perpendiculars are got by solving the equations:

3x+4y =12 and 4x-3y -2 = 0 simultaneously. From the former x = (12-4y)/3 . Substituting in the latter equation,

4(12-4y)/3-3y -2 = 0 Or

4(12-4y)-9y - 2=0. Or

48-25y-6 = 0 Or y = 42/25 And x =(12-4y)/3 =(12 -42*4/25)/3 = 44/25. Therefore the feet of perpendiculars are: (44/25,42/25)