Respuesta :
That point will be on the perpendicular line through the origin,
y = (-1/5)x
so will satisfy
(-1/5)x = 5x +4
(-26/5)x = 4
x = 4*(-5/26) = -10/13
y = (-1/5)x = 2/13
The point of interest is (-10/13, 2/13)
y = (-1/5)x
so will satisfy
(-1/5)x = 5x +4
(-26/5)x = 4
x = 4*(-5/26) = -10/13
y = (-1/5)x = 2/13
The point of interest is (-10/13, 2/13)
The point on the line [tex]y=5x+4[/tex] that is closest to the origin is [tex]\boxed{\left( { - \frac{{10}}{{13}},\frac{2}{{13}}} \right)}.[/tex]
Further explanation:
The formula for distance between the two points can be expressed as follows,
[tex]\boxed{{\text{Distance}} = \sqrt {{{\left( {{x_2} - {x_1}} \right)}^2} + {{\left( {{y_2} - {y_1}} \right)}^2}} }[/tex]
Given:
The line is [tex]y = 5x + 4.[/tex]
Explanation:
The coordinate of the origin is [tex]\left( {0,0} \right).[/tex]
The first point is [tex]\left( {x,y} \right)[/tex] and the second point is [tex]\left( {0,0} \right).[/tex]
The distance between the two points can be calculated as follows,
[tex]\begin{aligned}{\text{Distance}} &= \sqrt {{{\left( {x - 0} \right)}^2} + {{\left( {y - 0} \right)}^2}}\\ &= \sqrt {{x^2} + {{\left( {5x + 4} \right)}^2}}\\& = \sqrt {{x^2} + 25{x^2} + 40x + 16} \\&=\sqrt {26{x^2} + 40x + 16}\\\end{aligned}[/tex]
Differentiate the above equation with respect to x.
[tex]\begin{aligned}\frac{d}{{dx}}\left( {{\text{Distance}}} \right) &= \frac{d}{{dx}}\left( {\sqrt {26{x^2} + 40x + 16} } \right) \\& = \frac{1}{{2 \times \sqrt {26{x^2} + 40x + 16} }} \times \left( {52x + 40} \right) \\&= \frac{{\left( {52x + 40} \right)}}{{2 \times \sqrt {26{x^2} + 40x + 16} }} \\\end{aligned}[/tex]
Substitute the first derivative equal to zero.
[tex]\begin{aligned}\frac{d}{{dx}}\left( {{\text{Distance}}} \right) &= 0 \\ \frac{{\left( {52x + 40} \right)}}{{2 \times \sqrt {26{x^2} + 40x + 16} }} &= 0\\\left( {52x + 40} \right)&= 0\\52x &= - 40\\x &= \frac{{ - 40}}{{52}}\\x &= - \frac{{10}}{{13}}\\\end{aligned}[/tex]
Substitute x = - \frac{{10}}{{13}} in equation y=5x+4 to obtain the value of y.
[tex]\begin{aligned}y&= 5 \times \left({ - \frac{{10}}{{13}}} \right) + 4\\&=- \frac{{50}}{{13}} + 4\\&=\frac{{ - 50 + 52}}{{13}}\\&=\frac{2}{{13}}\\\end{aligned}[/tex]
The point on the line [tex]y=5x+4[/tex] that is closest to the origin is [tex]\boxed{\left( { - \frac{{10}}{{13}},\frac{2}{{13}}} \right)}.[/tex]
Learn more:
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Answer details:
Grade: High School
Subject: Mathematics
Chapter: Application of derivatives
Keywords: Derivative, attains, maximum, value of x, function, differentiate, minimum value, closest point, line, y=5x+4.
