line d y=2x+4 ,e y=2x+c find c and the lines are parallel and have 6 units between them
There are two lines that satisfy the given parameters, one above and to the left of
line d, and one below and to the right of line d. To find a point 6 units away from
line d, we need a line that is perpendicular to d. We get the equation of that line
by using the negative reciprocal of the m component, 2: that gives us -1/2. The
equation of this line is y = -1/2 x + 4, using the same y intercept. A segment of
that line that is 6 units represents the hypotenuse of a right triangle. If we are
looking at the segment that extends to the left of the given y intercept, we
consider the distance to be negative (-6). If we are looking at the segment that
extends to the right of the given y intercept, we consider the distance to be
positive (+6). To find the x and y co-ordinates, we use the sine and the cosine
of the angle (we'll call it a) formed by this new line and the x axis. The slope
m of a line is the y distance divided by the x distance. That is also the
definition of the tangent of the angle. Using the inverse tangent, we determine
that the angle is tan^-1 (-1/2) = -26.565 degrees. We need to keep all the
signs straight in order to get the correct values.
The sine of -26.565 degrees is -0.4472.
The cosine of -26.565 degrees is 0.8944.
y1 / -6 = sin a
y1 = -6 sin a = -6 * -0.4472 = 2.6832
This is measured from a horizontal line through the y intercept, because we are
constructing a right-triangle with one corner at the y intercept, so the point's
y co-ordinate is actually y = 2.6832 + 4 = 6.6832
x / -6 = cos a
x = -6 * 0.8944 = -5.3664
Making a quick check using x^2 + y^2 = r^2:
(-5.3664)^2 + 2.6832^2 = 28.79825 + 7.19956 = 35.99781 -6^2 = 36
Close enough considering the rounding in the calculations.
We have co-ordinate (-5.3664, 6.6832) lying 6 units from line d. Substituting
those values into y = 2x + c we have 6.6832 = 2 * (-5.3664) + c
6.6832 = -10.7328 + c
6.6832 + 10.7328 = c = 17.416
The equation of a line 6 units away from d and parallel to it, located above it,
is y = 2x + 17.416 <<<<<<<<
Working in the other direction, on the line below and to the right of line d,
y2 / 6 = sin a
y2 = 6 * sin a = 6 * -0.4472 = -2.6832 (that is, 2.6832 below d's y intercept)
y = 4 - 2.6832 = 1.3168
x / 6 = cos a
x = 6 * cos a = 6 * 0.8944 = 5.3664
The co-ordinates for this point, (5.3664, 1.3168), when substituted into
the equation y = 2x + f (f is a new y intercept) gives
1.3168 = 2 * 5.3664 + f
1.3168 - 10.7328 = f = -9.416
The equation of a line 6 units away from d and parallel to it, located below it,
is y = 2x - 9.416 <<<<<<<<