Write the equation of the family of lines with slope -1/2 and find the two members of the family 5 units away from the origin.
Let our two lines be y = -1/2 x ± k
The line y=2x is perpendicular to the given family of lines.
Suppose y=2x meets y = -1/2 x + k at (a,b).
Since b=2a, and a^2+b^2=25, find the intersection of the lines (solve for k) and now your lines are identified.
Why did the line go to therapy? It had a slope problem!
The equation of the family of lines with slope -1/2 is y = (-1/2)x + b. To find the two members of the family 5 units away from the origin, we substitute the x and y coordinates of the points into the equation and solve for b.
For the first point, (x, y) = (5, 0):
0 = (-1/2)(5) + b
0 = -5/2 + b
b = 5/2
So, the equation of the first line is y = (-1/2)x + 5/2.
For the second point, (x, y) = (0, 5):
5 = (-1/2)(0) + b
5 = b
Therefore, the equation of the second line is y = (-1/2)x + 5.
Remember, don't take these lines too personally—they might just have a slope issue!
To find the equation of a family of lines with a given slope, we use the point-slope form of a linear equation. The point-slope form is given by:
y - y₁ = m(x - x₁)
Where (x₁, y₁) is a point on the line, and m is the slope.
In this case, we are given that the slope is -1/2. Let's choose the point (x₁, y₁) as the origin (0, 0). Substituting these values into the point-slope form, we have:
y - 0 = -(1/2)(x - 0)
y = -(1/2)x
So, the equation of the family of lines with a slope of -1/2 is y = -(1/2)x.
To find the two members of the family that are 5 units away from the origin, we can substitute the x-coordinate, which is either 5 or -5, into the equation y = -(1/2)x.
For x = 5:
y = -(1/2)(5)
y = -5/2 = -2.5
So, one member of the family of lines that is 5 units away from the origin is y = -2.5 when x = 5.
For x = -5:
y = -(1/2)(-5)
y = 5/2 = 2.5
So, another member of the family of lines that is 5 units away from the origin is y = 2.5 when x = -5.
To find the equation of the family of lines with a given slope, we can use the point-slope form of a line equation, which is:
y - y₁ = m(x - x₁),
where (x₁, y₁) represents a point on the line, and m is the slope of the line.
In this case, the given slope is -1/2.
To find the two members of the family of lines that are 5 units away from the origin, we need to find two points on the lines that are 5 units away from the origin. Let's call these points (x₁, y₁) and (x₂, y₂).
Since the distance from the origin to a point (x, y) can be found using the Pythagorean theorem as:
√((x - 0)² + (y - 0)²) = √(x² + y²),
we can set up the following equation using the points (x₁, y₁) and (x₂, y₂):
√(x₁² + y₁²) = 5 ------(1)
√(x₂² + y₂²) = 5 ------(2)
Simplifying equations (1) and (2), we obtain:
x₁² + y₁² = 25 ------(3)
x₂² + y₂² = 25 ------(4)
Now let's find the two members of the family of lines:
1. Substitute the given slope into the point-slope form equation:
y - y₁ = (-1/2)(x - x₁) ------(5)
2. Rearrange equation (5) to find y in terms of x and x₁:
y = (-1/2)x + (1/2)x₁ + y₁ ------(6)
Now we have the equation of the family of lines with slope -1/2. To find two members of the family 5 units away from the origin, we can substitute the values of x₁ and y₁ from equations (3) and (4) into equation (6). This will give us two specific equations:
1st member of the family:
y = (-1/2)x + (1/2)x₁ + y₁ ------ (Substituting x₁ and y₁ from equations (3) and (4))
2nd member of the family:
y = (-1/2)x + (1/2)x₂ + y₂ ------ (Substituting x₂ and y₂ from equations (3) and (4))
With these substitutions, the specific equations of the two members of the family of lines that are 5 units away from the origin can be determined.