Question

Can someone please help me find the point on the line 2x+3y+4=0 which is closest to the point (3,-2).

Answer 1

Ok This is going to sound weird but when you write any point and a line in a piece of paper and try to find the point of the line where it's closest to that point what happens?

Answer 2

Draw a line between the two points.

Answer 3

a little confused

Answer 4

do you use the distance forula and the point given then somehow get another point plug in and take derivative to maximize?

Answer 5

\[2x+3y+4=0 \]closest to \((3,-2)\) right?

Answer 6

yes

Answer 7

ok first lets solve for y in the line to get \(y=-\frac{2}{3}x-\frac{4}{3}\) so any point on the line will look like \((x, -\frac{2}{3}x-\frac{4}{3})\)

Answer 8

then distance squared between a point on the line and \((3,-2)\) will be
\[(x-3)^2+(-\frac{2}{3}x-\frac{4}{3}+2)^2\]

Answer 9

thats as far as i got and then i got stuck

Answer 10

ok we have
\[(x-3)^2+(-\frac{1}{2}x-\frac{2}{3})^2\] now square and combine like terms first

Answer 11

my god is this ugly.
where did this problem come from? maybe there is an easier way, but once we have come this far might as well finish

Answer 12

yea its definitely not an eye pleasing problem in terms of integers

Answer 13

then we can do it the easier way.
looks like we get
\[\frac{5}{4}x^2-\frac{16}{3}x+\frac{85}{9}\]
http://www.wolframalpha.com/input/?i=%28x-3%29^2%2B%28-x%2F2-2%2F3%29^2

Answer 14

derivative is
\[\frac{5}{2}x-\frac{16}{3}\] set equal zero and solve for x

Answer 15

so quadratic formula im guessing

Answer 16

i get
\[x=\frac{32}{15}\]

Answer 17

no not quadratic formula, it is a line

Answer 18

kk let me check

Answer 19

I got 31/13 doing something completely different :P

Answer 20

wolfram gives the same answer

Answer 21

oh damn damn damn i made a mistake hold on!

Answer 22

ok its correct 31/13 , 2.923 are the cordinates thanks guys

Answer 23

i wrote it in wrong, sorry. let me try again
square of the distance is
\[\frac{13}{9}x^2-\frac{46}{9}x+\frac{85}{9}\]

Answer 24

sorry -2.923

Answer 25

derivative is
\[\frac{26}{9}x-\frac{46}{9}\] set = 0 and solve get
\[x=\frac{46}{26}=\frac{23}{13}\] did i make another mistake?

Answer 26

yes i see it

Answer 27

I'm not sure but what I did was make a function using point slope. I used the point that was given and the inverse slope of the function given. Used that function set it equal to the given function and found the point in where both cross each other.

Answer 28

should be
\[(x-3)^2+(-\frac{2}{3}x+\frac{2}{3})^2\]

Answer 29

@Romero yes that is the best method. slope will be perpendicular

Answer 30

There is always one way to solve it!! :)

Answer 31

slope of this line is
\[-\frac{2}{3}\] slope of perpendicular line is
\[\frac{3}{2}\] equaton of the line is
\[y+2=\frac{3}{2}(x-3)\] or
\[y=\frac{3}{2}x-\frac{5}{2}\] and then find the point of intersection