Find the point on the line 2 x + 6 y + 2 = 0 which is closest to the point ( 2, 0 ).
To find the point on the given line which is closest to the point (2, 0), we can use the concept of perpendicular distance. Follow these steps:
Step 1: Write the equation of the given line in slope-intercept form.
To do this, we need to solve the equation 2x + 6y + 2 = 0 for y. Rearranging the equation, we have:
6y = -2x - 2
y = (-2x - 2) / 6
Simplifying further:
y = (-1/3)x - 1/3
Step 2: Recall that the shortest distance between a point and a line is along the line perpendicular to it.
To find the slope of the line perpendicular to y = (-1/3)x - 1/3, we use the fact that perpendicular lines have slopes which are negative reciprocals.
The slope of the given line is -1/3, so the slope of the line perpendicular to it is 3 (negative reciprocal).
Step 3: Using the point-slope form of a line, plug in the values from the given point (2, 0) and the perpendicular slope (3) to find the equation of the line perpendicular to the given line and passing through (2, 0).
y - y₁ = m(x - x₁)
where (x₁, y₁) is the given point and m is the slope.
y - 0 = 3(x - 2)
y = 3x - 6
Step 4: Solve the two equations to find the intersection point.
We now have a system of equations:
y = (-1/3)x - 1/3
y = 3x - 6
To solve this system of equations, we can set the right-hand sides of the equations equal to each other:
(-1/3)x - 1/3 = 3x - 6
Multiply every term in the equation by 3 to eliminate the fractions:
-3x - 1 = 9x - 18
Rearrange and combine like terms:
12x = 17
Divide by 12:
x = 17/12
Substitute the value of x back into one of the original equations to find y:
y = (-1/3)(17/12) - 1/3
y = -17/36 - 3/12
y = -17/36 - 9/36
y = -26/36
Simplifying:
y = -13/18
Therefore, the point on the line 2x + 6y + 2 = 0 closest to the point (2, 0) is (17/12, -13/18).