To find the point on the line y = 3x + 5 that is closest to the origin (0, 0), we can use the concept of distance in coordinate geometry.
The distance D from any point (x, y) on the line to the origin is given by the distance formula:
D = √(x2 + y2)
Since y can be expressed in terms of x from the equation of the line, we substitute y = 3x + 5 into the distance formula:
D = √(x2 + (3x + 5)2)
To simplify the expression, we square both sides to eliminate the square root, which gives:
D2 = x2 + (3x + 5)2
Next, we can expand the equation:
D2 = x2 + (9x2 + 30x + 25)
This simplifies to:
D2 = 10x2 + 30x + 25
To minimize the distance, we need to find the vertex of this quadratic equation, as it will give us the minimum distance. The x-coordinate of the vertex can be calculated using the formula:
x = -b / 2a
In our case, a = 10 and b = 30. Plugging in the values:
x = -30 / (2 * 10) = -1.5
Now that we have x, we can find y by substituting back into the original line equation:
y = 3(-1.5) + 5 = -4.5 + 5 = 0.5
Finally, the point on the line y = 3x + 5 that is closest to the origin is: (-1.5, 0.5).