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A line has slope 2. Find the area of the triangle formed by this line and the coordinate axes, if the distance between the origin and the line is 5

All Help is appreciated!

 Dec 3, 2023
 #1
avatar+222 
-2

Any line with slope 2 can be rewritten in the form y=2x+b for some constant b. Since the distance between the origin and the line is 5, we have that 5=22+12​⋅∣b∣, so ∣b∣=355​​. There are two possible lines with slope 2 and distance 5 from the origin: y=2x−355​​ and y=2x+355​​. These lines intersect at the point (5*sqrt(5)/15, 5*sqrt(5)/3).

 

Since the slope of the line is 2, the line passes through the point (0,5*sqrt(5)/3​​). The area of the triangle formed by the line and the coordinate axes is 25*sqrt(5)/6​​​ square units.

 Dec 4, 2023
 #2
avatar+36900 
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The PERPINDICULAR line from the origin has a slope of   - 1/m =  -1/2 

  so  this line is  y = - 1/2 x 

   A circle of radius 5 , centered at the origin (  x^2 + y^2 = 5^2 ) 

    Intersects are prpindicular line at   (4.472 , -2.236) 

        NOW....the equation of the line that contains THIS point and has a slope of 2 is 

                     y + 2.236 = 2 ( x - 4.472) 

                   or  y = 2x - 11.18 

 

This has a y-axis intercept of  - 11.18    and the x intercept is 5.59 

Area of the triangle is 1/2 base * height     

     = 1/2   * 11.18  * 5.59 =  31.25 units^2 

Here is a Desmos graph showing all of this :

 Dec 5, 2023

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