Question

if the foot of the perpendicular from (-4, 5) to the straight line 3x-4y-18=0 is ( alfa, beta) then the value of alfa+ beta is​

Answer 1

Step-by-step explanation:

the graph looks something like this, now note that (a,b) is the point that lies on the intersection of both the lines.

first construct a line equation for the line joining (a,b) and (-4,5) using two point formula

x-(-4)=((b-5)/(a+4))(y-5)

which on simplifying becomes

x(b-5)-y(a+4)=-5a-4b ...(i)

now slope of this is m1=(b-5)/(a+4) [m=-A/B]

now, (a,b) also falls on 3x-4y-14=0 so it must satisfy it as well, hence 3a-4b-14=0 ...(ii), its slope m2=3/4

as both of these lines are perpendicular to each other hence the product of their slopes must be equal to -1

so, m1×m2=-1 => 4a+3b=-1 ...(iii)

now, Solving equation (ii)&(iii) for a and b,

we get a=1.52 and b=-9.44

hence, (a,b)=(1.52,-9.44)