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

Solution:

The point on the line y = 4x + 5 that is closest to the origin will be on the line perpendicular to the origin.

y = 4x + 5 [Given]

Slope = m₁ = 4

When two lines are perpendicular to each other

m₁ . m₂ = - 1

m₂ = -1/m₁

m₂ = -1/4

y = -1/4 x

Substitute the value of y in the given equation

- 1/4 x = 4x + 5

- 1/4 x - 4x = 5

Taking LCM

(-1 - 16)/4 x = 5

-17/4 x = 5

By cross multiplication

x = 5 × -4/17

x = -20/17

y = - 1/4 (-20/17) = 5/17

Therefore, the point on the line closest to the origin is (-20/17, 5/17).

---

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

Summary:

The point on the line y = 4x + 5 that is closest to the origin is (-20/17, 5/17).