Find the point on the line y = 4x + 3 that is closest to the origin?

Solution:

The point on the line y = 4x + 3 that is closest to the origin will be on the line perpendicular to the origin.

y = 4x + 3 [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 + 3

- 1/4 x - 4x = 3

Taking LCM

(-1 - 16)/4 x = 3

-17/4 x = 3

By cross multiplication

x = 3 × -4/17

x = -12/17

y = - 1/4 (-12/17) = 3/17

Therefore, the point on the line closest to the origin is (-12/17, 3/17).

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Find the point on the line y = 4x + 3 that is closest to the origin?

Summary:

The point on the line y = 4x + 3 that is closest to the origin is (-12/17, 3/17).