Can u find the shortest distance from P( 5,4) to the line Q(3x+5y-4=0)?

i know all the steps...find the slope of the line....find the negative reciprocal....find Y-intercept. of PQ....find equation of PQ...use substitution/elimination to find the point fo intersection...then use the distance formula...


i just want the answer and show me the workin out....PLEASE

Can u find the shortest distance from P( 5,4) to the line Q(3x+5y-4=0)?
The shortest distance will be perpendicular to the line Q.


Gradient of Q is -3/5 so gradient of perpendicular is 5/3.


Line through P with gradient 5/3 is y - 4 = (5/3)*(x - 5) or


y = 4 + (5/3)(x - 5)


y = (5/3)x - (13/3)





Q's equation can be rewritten as y = (-3/5)x + (4/5)


Line through P hits Q when


y = (5/3)x - (13/3) = (-3/5)x + (4/5)


25x - 65 = -9x + 12


34x = 77


x = 77/34


This gives y = (5/3)*(77/34) - (13/3) = -57/102





Distance between (5, 4) and (77/34, -57/102)


= sqrt [ (5 - 77/34)^2 + (4 + 57/102)^2 ] = 5.32 (2dp)





The person who made up the question could have designed it to give a nicer point of intersection. Is this why you thought you weren't doing it correctly?
Reply:to do this, you must know that the shortest distance between any two points is a straight line. So the shortest distance between a point and a line is the perpendicular.





the first step is to get the line in a usable form.





3x + 5y - 4 = 0 (subract 3x, add 4)


5y = -3x + 4 (divide by 5)








y = (-3/5)x + 4/5





slope (m) = -3/5





next, we find the equation of the line that passes through (5,4) and is perpendicular to line Q.





a perpendicular line will have the opposite reciprocal slope of the original line.





-1 / (-3/5) = 5/3 ... the new slope is 5/3





y = mx+b


so the new equation is y = (5/3)x + b





plug in the values that the perpendicular line goes through (5,4), and solve for b.





4 = (5/4)(5) +b





b = 25/4 - 4





y = (5/3)x + 25/4 - 4





now set the two equations equal to each other to find x.





(5/3)x + 25/4 - 4 = (-3/5)x + 4/5





(5/3)x + (3/5)x = (4/5) - (25/4) + 4...find common denom.





(25/15)x + (9/15)x = (4/5) - (25/4) + 4





(34/15)x = (4/5) - (25/4) + 4 ...using a calculator, because im too lazy to find x using algebra.





there could be a mistake, cause x = -87/136 or -.6397





y = -(3/5)(-.6397) + 4/5


y = 1.1838





It's y-y/x-x to find distance, i think so





(4 - 1.1838) / (5 - (-.6397)) = .6767 (approx.)





answer is .6767





i hope all the work is right, sorry if it isn't.