What is the point on the line y = 2x + 3 that is closest to the origin?

Solution:

The point on line y = 2x + 3 closest to the origin is the point of intersection of y = 2x +3 and line perpendicular to it passing through (0, 0).

We know that equation of line perpendicular to Ax + By + C = 0 and passing through (x₁, y₁) is

B(x - x₁) - A(y - y₁) = 0.

Here, A = 2, B = -1 and (x₁, y₁) = (0, 0)

⇒ -1(x - 0) -2(y - 0) = 0

⇒ x + 2y = 0 --- (1)

Now, the nearest point on line y = 2x + 3 from the origin is the point of intersection of 2x - y + 3 = 0 and x + 2y = 0.

Solving, x + 3(3x + 4) = 0

⇒ 10x + 12 = 0

⇒ x = -12/10

⇒ x = -6/5

Substitute the x = -6/5 in equation(1)

⇒ -6/5 + 2y = 0

⇒ y = 3/5

Therefore, the required point is (-6/5, 3/5)

What is the point on the line y = 2x + 3 that is closest to the origin?

Summary:

The point on the line y = 2x + 3 that is closest to the origin is (-6/5, 3/5).