find the distance between lines whose equations are: y=-1/4x+2 and y=-1/4x-9/4.
d=�ã(1-(-3)^2 +(1-2)^2
I get
|2+9/4|/√(1/16 + 1)
= (17/4)/√(17/4)
= √17/2
To see how, take a look at
http://en.wikipedia.org/wiki/Distance_between_two_straight_lines
It seems like there might be a mistake in the formula you provided. The formula to find the distance between two points is actually d = sqrt((x2 - x1)^2 + (y2 - y1)^2). However, since you mentioned lines, I assume you want to find the distance between the parallel lines represented by the given equations.
In this case, they both have the same slope (-1/4), which means they are parallel. Parallel lines never intersect, so the distance between them is constant. To find it, we can choose any point on one line and calculate the perpendicular distance to the other line.
Let's pick a point on the line y = -1/4x + 2. For simplicity, let x = 0, so the point is (0, 2). Now, using the formula for the distance between a point and a line, we have:
d = |(-1/4)(0) + (1)(2) + (9/4)| / sqrt((-1/4)^2 + 1^2)
= |0 + 2 + 9/4| / sqrt(1/16 + 1)
= |2 + 9/4| / sqrt(1/16 + 1)
= |17/4| / sqrt(1/16 + 1)
= 17/4 / sqrt(17/16)
= 17/4 / (sqrt(17)/4)
= 17 / sqrt(17)
So, the distance between the given parallel lines is 17 / sqrt(17).
To find the distance between two lines, we can use the formula:
d = |(c2 - c1)/(√(a^2 + b^2))|
where a, b are the coefficients of x in the equations of the lines, and c1, c2 are the constants on the right side of the equations.
For the first line y = -1/4x + 2, we have:
a1 = -1/4, b1 = 1, c1 = 2
For the second line y = -1/4x - 9/4, we have:
a2 = -1/4, b2 = 1, c2 = -9/4
Plugging these values into the formula, we have:
d = |((-9/4) - 2)/(√((-1/4)^2 + 1^2))|
Simplifying further:
d = |-17/4/(√(1/16 + 1))|
d = |-17/4/(√(17/16))|
Simplifying the denominator:
d = |-17/4/(√(17)/4)|
Multiplying by the reciprocal of the denominator:
d = |-17/(√17)|
So, the distance between the two lines is | -17/(√17) | or 17/(√17).
To find the distance between two lines, you can use the formula for the distance between a point and a line. In this case, we need to find the distance between the lines represented by the equations y = -1/4x + 2 and y = -1/4x - 9/4.
First, we need to find the point on one line that is closest to the other line. To do this, we can set the two equations equal to each other and solve for x:
-1/4x + 2 = -1/4x - 9/4
Simplifying this equation, we get:
9/4 = 0
This equation has no solution, which means the two lines are parallel and will never intersect.
Since the lines are parallel, the distance between them will be constant. To find this distance, we can pick any point on one of the lines, calculate the perpendicular distance from that point to the other line, and divide it by the square root of the sum of the squares of the coefficients of x and y in the second equation.
Let's choose a point (x0, y0) on the line y = -1/4x + 2. In this case, we can use the y-intercept (0, 2). The equation of the second line can be rewritten as y = -1/4x - 2 1/4.
The perpendicular distance (d) between the two lines can be calculated using the formula:
d = |(-1/4)(x0) - (-1/4)(x0) - 2 1/4 + 2| / sqrt((-1/4)^2 + 1^2)
Substituting the values, we get:
d = |(-1/4)(0) - (-1/4)(0) - 2 1/4 + 2| / sqrt((-1/4)^2 + 1^2)
= |0 - 0 - 2 1/4 + 2| / sqrt((1/16) + 1)
= |0 - 2 1/4 + 2| / sqrt(17/16)
= |-2 1/4 + 2| / sqrt(17/16)
= |1/4| / sqrt(17/16)
= 1/4 / sqrt(17/16)
= 1/4 * sqrt(16/17)
= sqrt(16) / (4 * sqrt(17))
= 4 / (4 * sqrt(17))
= 1 / sqrt(17)
Therefore, the distance between the lines y = -1/4x + 2 and y = -1/4x - 9/4 is 1 / sqrt(17).