SOLUTION: Analytic Geometry 11. Determine the shortest distance from the origin to the line represented by y=1/2x-2. Can u please help me ???????????? thanksss

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Question 200407: Analytic Geometry
11. Determine the shortest distance from the origin to the line represented by y=1/2x-2.
Can u please help me ????????????
thanksss
Found 2 solutions by vleith, Alan3354:
Answer by vleith(2983) About Me  (Show Source):
You can put this solution on YOUR website!
Draw the line.
Then imagine a small circle centered at the origin. As that circle radius grows, the circle will eventually touch the line. The point where it 'just touches' results in the drawn line being a tangent to the circle.
What do you know about the radius of a circle at a tangent point and a tangent to a circle at that point? They are perpendicular.
What do you know about the slopes of lines that are perpendicular? They are negative inverses.
So find the line that has a slope of -2 and contains the point (0,0).
Solved by pluggable solver: FIND a line by slope and one point

What we know about the line whose equation we are trying to find out:

  • it goes through point (0, 0)

  • it has a slope of -2



First, let's draw a diagram of the coordinate system with point (0, 0) plotted with a little blue dot:



Write this down: the formula for the equation, given point x%5B1%5D%2C+y%5B1%5D and intercept a, is

y=ax+%2B+%28y%5B1%5D-a%2Ax%5B1%5D%29 (see a paragraph below explaining why this formula is correct)

Given that a=-2, and system%28+x%5B1%5D+=+0%2C+y%5B1%5D+=+0+%29+, we have the equation of the line:

y=-2%2Ax+%2B+0

Explanation: Why did we use formula y=ax+%2B+%28y%5B1%5D+-+a%2Ax%5B1%5D%29 ? Explanation goes here. We are trying to find equation y=ax+b. The value of slope (a) is already given to us. We need to find b. If a point (x%5B1%5D, y%5B1%5D) lies on the line, it means that it satisfies the equation of the line. So, our equation holds for (x%5B1%5D, y%5B1%5D): y%5B1%5D+=+a%2Ax%5B1%5D%2Bb Here, we know a, x%5B1%5D, and y%5B1%5D, and do not know b. It is easy to find out: b=y%5B1%5D-a%2Ax%5B1%5D. So, then, the equation of the line is: +y=ax%2B%28y%5B1%5D-a%2Ax%5B1%5D%29+.

Here's the graph:



y+=+-2x
Now that you have the equations for two lines, solve them to find the point they have in common.
y+=+%281%2F2%29x+-+2
y+=+-2x
-2x+=+%281%2F2%29x+-+2
-%285%2F2%29x+=+-2
5x+=+4
x+=+4%2F5
y+=+-8%2F5

Now you have two points, the origin (0,0) and the point of intersection (0.8,-1.6). Find the length between them and you have your answer.
Solved by pluggable solver: Distance between two points in two dimensions
The distance (denoted by d) between two points in two dimensions is given by the following formula:

d=sqrt%28%28x1-x2%29%5E2+%2B+%28y1-y2%29%5E2%29

Thus in our case, the required distance is
d=sqrt%28%280-0.8%29%5E2+%2B+%280--1.6%29%5E2%29=+1.78885438199983+


For more on this concept, refer to Distance formula.


graph%28400%2C400%2C-5%2C5%2C-5%2C5%2C+0.5x-2%2C+-2x%29

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Determine the shortest distance from the origin to the line represented by y=1/2x-2.
----------------
The shortest distance from a point to a line is along the line perpendicular.
The slope of y = (1/2)x - 2 is 1/2.
The line perpendicular has a slope that's the negative inverse, m = -2.
y-y1 = m(x-x1) is the eqn of the perpendicular line, where (x1,y1) is (0,0)
y = -2x
-------
Solve for x at the intersection:
-2x = x/2 - 2
-4x = x - 4
x = 0.8
y = -1.6
-------
The distance is sqrt(0.8^2 + 1.6^2)
= sqrt(3.2)
= 0.8*sqrt(5)
= ~1.7889