Find the point on the line y = 5x + 4 that is closest to the origin?

Solution:

The point on line y = 5x + 4 closest to the origin is the point of intersection of y = 5x + 4 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 = 5, B = -1 and (x₁, y₁) = (0, 0)

⇒ -1(x - 0) - 5(y - 0) = 0

⇒ x + 5y = 0 --- (1)

Now, the nearest point, say P, on line y = 5x + 4 from the origin is the point of intersection of 5x - y + 4 = 0 and x + 5y = 0.

Solving, x + 5(5x + 4) = 0

⇒ 26x + 20 = 0

⇒ x = -20/26

⇒ x = -10/13

Substituting x = -10/13 in equation (1), we get

⇒ -10/13 + 5y = 0

⇒ y = 2/13

Therefore, the required point is (-10/13, 2/13)

Find the point on the line y = 5x + 4 that is closest to the origin?

Summary:

The point on the line y = 5x + 4 that is closest to the origin is (-10/13, 2/13).