The axes being inclined at an angle of 30°, find the equation to the straight line
which passes through the point(-2,3) and is perpendicular to the straight line
y + 3x = 6
given line 3x+y=6.
slope=-3.slope of another line is 1/3.
slope point equation (y-Y)=m(x-X).
given point (-2,3).
y-3=1/3(x+2).
3y-9=x+2.
x-3y+11=0
To find the equation of a line that is perpendicular to another line, we need to determine the slope of the given line and then find the negative reciprocal of that slope.
The equation of the given line is y + 3x = 6. We can rearrange it to the slope-intercept form, y = mx + b, where m is the slope.
Let's rearrange the equation to solve for y:
y + 3x = 6
y = 6 - 3x
Comparing this equation to y = mx + b, we see that the slope (m) of the given line is -3.
The negative reciprocal of -3 is 1/3. So, the slope of the line perpendicular to the given line is 1/3.
Since the given line is inclined at an angle of 30°, the axes are inclined at an angle of 30° as well. This means that the given line is inclined 60° with respect to the positive x-axis.
To find the slope of the line in terms of the inclined axes, we need to adjust the slope by taking into account the angle of inclination. The formula for converting from the inclined axes to the Cartesian axes is:
slope_inclined_axes = tan(θ) * slope_Cartesian_axes
where θ is the angle of inclination, which is 30° in this case.
Let's calculate the slope in terms of the inclined axes:
slope_inclined_axes = tan(30°) * (1/3)
slope_inclined_axes = (√3/3) * (1/3)
slope_inclined_axes = √3/9
Now we can use the point-slope formula to find the equation of the line perpendicular to the given line that passes through the point (-2, 3).
The point-slope form of a line is: y - y1 = m(x - x1), where (x1, y1) is a point on the line and m is the slope.
Plugging in (-2, 3) as (x1, y1) and √3/9 as the slope (m):
y - 3 = (√3/9)(x - (-2))
y - 3 = (√3/9)(x + 2)
This is the equation of the straight line that passes through (-2, 3) and is perpendicular to the line y + 3x = 6.