Find the point on the line y = 4x+2 that is closest to the origin.

Solution:

The point on the line y = 4x + 2 which is closest to the origin will be on the line perpendicular to the origin.

y = 4x + 2 [Given]

Slope m₁ = 4

We know that when two lines are perpendicular to each other

m₁ . m₂ = -1

So we get

m₂ = -1 / m₁

m₂ = -1/4

y = -1/4 x

Let us substitute the value of y in the equation given

-1/4 x = 4x + 2

-1/4 x - 4x = 2

By taking LCM

(-1 - 16)/4 x = 2

-17/4 x = 2

Now by cross multiplying

x = 2 x -4/17

x = -8/17

Substitute the value of x

y = -1/4 (-8/17) = 2/17

Therefore, the point closest to the origin is (-8/17, 2/17).

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Find the point on the line y = 4x+2 that is closest to the origin.

Summary:

The point on the line y = 4x+2 that is closest to the origin is (-8/17, 2/17).