Find the distance between the point (0, 0) ) and the line with the - 2x - y = - 3

Answer 1

there are different approaches to the problem

you want the perpendicular distance from the origin to the line

first find the equation of the line perpendicular to the original line that goes through the origin

you have -2x-y=-3

2x+y=3

y=-2x+3

the slope of the line perpendicular to this line is the negative reciprocal of the slope of the original line

m=1/2

also, the line goes through the origin so its y-intercept is 0

y=(1/2)x is the equation of the line perpendicular to the original line

y=-2x+3

y=(1/2)x

solve this system for the point of intersection

subtract the equations

0=(-2 1/2)x+3

(2 1/2)x=3

(5/2)x=3

x=6/5

substitute x=6/5 to find y

y=(1/2)(6/5)

y=3/5

the point of intersection is (6/5, 3,5)

now find the distance from (0,0) to (6/5, 3/5) using the distance formula

√((6/5)^2+(3/5)^2)=

√(36/25 + 9/25)=

√45/25=

√9/5=

3/√5=

3√5/5 is the perpendicular distance from the origin to the original line

Answer 2

Hi Abigail,

The distance from a point to a line is defined as the segment length from the point perpendicular to the line.

The given equation of the line: -2x - y = -3.

Solve this for y.

y = -2x + 3. It's slope is -2. The slope of the line perpendicular to the given line is (1/2).

The origin (0, 0) is a point on that perpendicular line: it's equation: y = (1/2)x.

Now find the intersection of the 2 lines.

Since y = -2x + 3 and y = (1/2) x, we set the right sides of the equations equal to each other.

-2x + 3 = (1/2)x

3 = (5/2)x

x = (6/5), then y = (3/5).

Now use the distance formula to find the distance between the origin and

((6/5), (3/5)).

The result is (3√5/5).

Regards,

Anita A.

Answer 3

Hello Abigail,

For finding the distance between a line and a point, you need to know a formula:

d = | a * x₁ + b* y₁+c | / ( a² + b² )^1/2

point is (0,0) = (x_{1 ,} y₁)

you need to write an equation of line such as -2x -y +3 =0 then recognize a = -2, b = -1, c = 3

Now you replace all information in formula

d = | -2*0 -1*0 +3 | / ( 4 + 1 )^1/2

d = |3| / 5^1/2

d = 3 * 5^1/2/ 5