Question

the point on the line −2x+3y+10=0 which is closest to the point (5 , -4) has x coodinate and y coordinate of what?

I cant seem to get this question, I know you have to use the distance formula, and express one of the variables as the other, but I cant seem to get it? Any help please!

Answer 1

the vector perpendicular to the line is (-2,3) --this can be directly taken from the coefficients of x and y in the line expression.
We can find the parametric equation of the line that contains the given point (5,-4) as follows:

<x,y>=<5,-4>+t <-2,3>
or
x=5 - 2t
y=-4+3t

Now find the point of intersection of the original and this perpendicular line:
-2x+3y+10=0 --> plug in the above x and y :
-10+4t+9t-12+10=0 ---> solve for t:
t=12/13 ---> find the corresponding coordinates (x*,y*) of intersect
x*=41/13
y*=-16/13
Answer: (41/13, -16/13)

have a good one