Question

A line goes through the points (2, 5) and (6, −1). Let P be the point on this line that is closest to the origin. Calculate the coordinates of P.

Answer 1

@jhonyy9

Answer 2

Find first the equation of the line going thru the 2 points

Answer 3

You will find it
\[
y=8-\frac{3 x}{2}
\]

Answer 4

Write it as
\[ 3 x + 2y -16=0\]

Answer 5

Second, find a line that's perpendicular to it, that also goes through the origin.

Answer 6

Use now the distance formula from a point\( (m,n)\) to a line
a x + by + c=0

Answer 7

\[
q=\frac {|a m + b n + c|}
{\sqrt{a^2+b^2}
}

\]

Answer 8

Here's what I thought...

It seems to be like the x-intercept or the y-intercept might be the closer points to the origin. I calculated the x-int and the y-int and saw that (4.5,0) <- is the closer one

Answer 9

replace m=0, n=0, a=3, b=2, c=-16 in the above to get your answer

Answer 10

But I am not so sure whether that is exactly correct or not

Answer 11

You can also do it by the perpendicular line

Answer 12

LIke I said ^^

THe shortest distance is basically a radius, and the line is a tangent.

Answer 13

I searched through some websites and they said that if I find the perpendicular line that passes through the origin and the point of intersection of the original line and the perpendicular line will give me the point. Is it correct to assume that?

Answer 14

\[
\frac {16}{ \sqrt{3^2+2^2}}=\frac {16}{\sqrt{13}}
\]

Answer 15

I just tried out the thing that I just said recently and my graphing calculator says that the point of intersection is (3.6,1.8)

Answer 16

Yes, that's what I said in my first post.

Answer 17

That's correct right--the coordinates that I just said?

Answer 18

The equation of the perpendicular line is y= 2 x/3
in intersects the original line at the point (48/13,32/13)

Answer 19

Find the distance from that point to the origin, you find my answer by the firstmethod

Answer 20

\[
\sqrt{\left(\frac{32}{13}\right)^2+\left(\frac{48}{13}\right)^2}=\frac{16}{\sqrt{13}}
\]

Answer 21

The equation of the original line I got is y=-2x+9
The perpendicular line that would cross through the origin would be y = 1/2x

Answer 22

No the original equation is
y =-3 x/2 +8

Answer 23

Recheck your computation

Answer 24

got it

Answer 25

So now I got approximately (3.69, 2.47)

Answer 26

Which I think is about what you got as well

Answer 27

Yes

Answer 28

Use the distance formula from (0,0) to (48/13,32/13)
to get your answer