How do you find the point on a line that is closest to the origin for this line?

Question

find the point on the line y=-3x+2 that is closest to the origin

Answer 1

The shortest distance from a point to a line is along the perpendicular to the line.and passing through the point.

The equation of the line perpendicular is y = (1/3)x

Determine the point of intersection

(1/3)x = -3x + 2

(10/3)x = 2

10x = 6

x = 0.6

y = 0.2

Use the distance formula (or Pythagoras) to answer

Answer 2

To find the point on the line closest to the origin, we are actually being asked to minimize distance. Since any point on the line would have coordinates ( x, -3x+2), the equation we will differentiate here will be the distance formula, using the above point and the point ( 0,0 ). The equation would be written as d = square root of [ x² + (-3x+2)² ], which simplifies to square root of ( 10x² - 12x + 4 ) or ( 10x² - 12x + 4)^1/2.

To differentiate this, we will need to use Chain Rule because it involves a composition of functions, so d' = 1/2( 10x² - 12x + 4)^-1/2( 20x - 12). Simplifying this, we have a fraction: d' = ( 10x - 6 )/ (10x² - 12x + 4)^-1/2. To find the minimum distance, we need to find where the derivative = 0. The only x value that will make this fraction = 0 is the value that makes the numerator = 0, and this would be x = 3/5. If we check x values on either side of 3/5 to confirm the minimum ( derivative should be negative to the left of 3/5 and positive to the right of 3/5), we see this is the case. We should also confirm that x = 3/5 does NOT make the denominator = 0, which it does not.

So the point on the line closest to the origin will have an x-coordinate of 3/5, or .6, and we insert this into the expression for y to find the y-coordinate, which would be .2, so the point on the line closest to the origin is ( .6, .2 ).