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)   (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 and intercept a, is

(see a paragraph below explaining why this formula is correct)

Given that a=-2, and , we have the equation of the line:



Explanation: Why did we use formula ? 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 (, ) lies on the line, it means that it satisfies the equation of the line. So, our equation holds for (, ): Here, we know a, , and , and do not know b. It is easy to find out: . So, then, the equation of the line is: .

Here's the graph:




Now that you have the equations for two lines, solve them to find the point they have in common.








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:



Thus in our case, the required distance is



For more on this concept, refer to Distance formula.
Answer by Alan3354(69443)   (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
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