A line in the coordinate plane has a slope of \(4\) and a distance of \(1\) unit from the origin. Find the area of the triangle determined by the line and the coordinate axes.
Thank you!
+36916 Slope = rise over run or y /x so y = 4x
Pythag theor x^2 + (4x)^2 = 1^2
17x^2 = 1
x^2 = 1/17
x = sqrt (1/17) y = 4 sqrt (1/17)
Area = 1/2 b h = 1/2 x y = 1/2 sqrt 1/17 * 4 sqrt 1/17
= 2 (1/17) = 2/17 units2
+128474 See the image below
Construct circle x^2 + y^2 = 1
Construct line y = (-1/4)x
Sub the linear equation into the circular equation to find the x intersection of line and the circle
x^2 +[ (-1/4)x]^2 = 1
x^2 + (1/16)x^2 = 1
(17/16)x^2 = 1
x^2 = 16/17 take the negative root
x= -4/√ 17
And y = (-1/4)(-4/√ 17) = 1/√ 17
The equation of the line tangent to this circle at this point will have the equation
y = 4 (x + 4/√ 17) + 1/√ 17
The intersection of this line and the x axis can be found as
0 = 4 ( x +4/√ 17) + 1/√ 17
-1/√ 17 = 4 ( x + 4/√ 17)
-1/ [ 4√ 17] = x + 4/√ 17
-1/[4√ 17] - 16/[4√ 17)
-17/ [ 4√ 17] =
-√ 17/4
And the intersection of this line with the y axis can be found as
y = 4[ 0 + 4/√ 17] + 1/√ 17 = 17/√ 17 = √ 17
So the area of this triangle =
(1/2) base * height
The base length = √ [ (√ 17)^2 + (-√ 17/4)^2 ] = √ [ 17 + 17/16] = √ [ 16 * 17 + 17] /4 = 17/4
So....the area of this triangle is
(1/2) (17/4) (1) = 17/8 units^2
+36916 Sorry..... I gave answer for a POINT that is one unit from the origin on a line with slope = 4
