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

SOMEONE ASKED πŸ‘‡

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

HERE THE ANSWERS πŸ‘‡

  • Hello,

    Every point M(x; y) of the line fits the equation:

    y = 4x + 3

    The distance from such a point M(x; y) to the origin would be:

    distance = √(x² + y²)

    Since y=4x+3, we would get:

    distance = √[x² + (4x + 3)²] = √(17x² + 24x + 9)

    Normally, to find the minimal value of this distance, we would then need to differentiate this expression versus x and find the value xβ‚€ which would make this derivative nil.

    However, if the distance is minimal when x=xβ‚€, it stands to reason that the square of that minimal distance when x=xxβ‚€ would also be minimal versus the square of all other x. So:

    (distance)Β² = 17xΒ² + 24x + 9

    [(distance)Β²]’ = 34x + 24

    This will be minimal for:

    34x + 24 = 0

    x = -24/34 = -12/17

    So xβ‚€ = -12/17

    And yβ‚€ = 4xβ‚€ + 3 = -48/17 + 51/17 = 3/17

    And the minimal distance will be:

    √(xβ‚€Β² + yβ‚€Β²) = √[(-12/17)Β² + (3/17)Β²] = (3√17)/17 β‰ˆ 0.728

    (xβ‚€; yβ‚€) = (-12/17; 3/17)

    Methodically,

    Dragon.Jade πŸ™‚

  • y = mx + b where b is the y-intercept (where x = 0)

    so 3 is the y-intercept

    that is close ( 0, 3 )

    Let’s check the x-intercept (where y = 0 )

    4x + 3 = 0

    4x = – 3

    x = – 0.75 that is closer than above, and y-value is zero, point (0.75, 0) *

    that looks likely to be the one closest to origin

    because if we go to x = -1

    y = -1 getting further away from (0, 0)

    but to be sure let’s try x = 0.5

    and y = 5 there, too far away, above origin

    but let’s try x = – 0.5

    then y = 1 Ζ’α΄€Κ€Ρ‚her than * above, new result is point (-0.5, 1)

    Let’s try – 0.25 for x

    then y = 2 closer still ( -0.25, 2) Ζ’α΄€Κ€Ρ‚her away, both values need to be closest to (0,0)

    Let’s try x = – 0.6

    then y = 0.6 giving (- 0.6, 0.6) closer than * above

    since we got closer, we’ll try – 0.55 for x

    then y = (-0.55)(4) + 3 = 0.8 getting further away

    to be sure we’ll try x = – 0.59

    then y = 0.64 (-0.59, 0.64)

    so the bear likes (-0.6, 0.6) best, BECAUSE a lower or higher value for x is Ζ’α΄€Κ€Ρ‚her

    Hmmmm, if Dragon head is closer, I should have tried lower value for y, which Dragon says is 3/17

    giving 3 = 68x + 51, 68x = – 48, x = – 48/68, = – 12/17 minimizing closest diagonal, a bit fussier than I thought wanted, but by my method 0.70588 seemed further away for x, my diagonal is 0.84853 and Dragon’s is closer, yes.

    Source(s): Needs careful checking, bear is not a math head.
  • (-1,-1)

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