To find the point on the line y = 3x + 4 that is closest to the origin (0,0), we can use a method involving distance minimization.
The distance D from any point (x, y) on the line to the origin can be calculated using the distance formula:
D = √(x2 + y2)
Since y is defined by the line equation, we can substitute:
D = √(x2 + (3x + 4)2)
Expanding the equation, we get:
D = √(x2 + (9x2 + 24x + 16))
Combining like terms:
D = √(10x2 + 24x + 16)
To minimize D, we can minimize D2 because it’s easier to work with without the square root:
D2 = 10x2 + 24x + 16
Now we take the derivative of D2 with respect to x, set it equal to zero, and solve for x:
f(x) = 10x2 + 24x + 16
f‘(x) = 20x + 24 = 0
Solving for x, we have:
20x = -24
x = -24/20 = -6/5
Now, substitute x back into the line equation to find y:
y = 3(-6/5) + 4 = -18/5 + 20/5 = 2/5
The coordinates of the point on the line
y = 3x + 4 that is closest to the origin are:
(-6/5, 2/5).