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!

itsash Dec 3, 2023

#1**-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.

BuiIderBoi Dec 4, 2023

#2**+1 **

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 :

ElectricPavlov Dec 5, 2023