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What is the smallest possible distance between origin and any point on

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What is the smallest possible distance between origin and any point on  [#permalink]

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New post 26 Mar 2018, 22:52
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Question Stats:

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What is the smallest possible distance between origin and any point on the line \(y=\frac{1}{2}x+50\)?

A. 25
B. \(20\sqrt{5}\)
C. 50
D. \(50\sqrt{2}\)
E. 100
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What is the smallest possible distance between origin and any point on  [#permalink]

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New post 27 Mar 2018, 06:53
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rishabhmishra wrote:
What is the smallest possible distance between origin and any point on the line \(y=\frac{1}{2}x+50\)?

A. 25
B. \(20\sqrt{5}\)
C. 50
D. \(50\sqrt{2}\)
E. 100


Answer is B
The Smallest distance between the origin and any point on the line \(y=\frac{1}{2}x+50\) would be length of perpendicular drawn
from origin to line \(y=\frac{1}{2}x+50\).

assume the equation of line passing through origin and perpendicular to line \(y=\frac{1}{2}x+50\) be y= mx +c.
as two line are perpendicular and non vertical, product of their slope would be -1.
so m = -2.
as (0,0) lies on y= mx +c, c should be zero.
So equation of perpendicular line to \(y=\frac{1}{2}x+50\) and passing through origin is y=-2x.
Point of intersection of these line would x= -20 and y = 40.

Distance of this point from origin = \(\sqrt{[(-20-0)^2 +(40-0)^2]}\) =20\(\sqrt{5}\)
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What is the smallest possible distance between origin and any point on  [#permalink]

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New post 29 Mar 2018, 01:08
1
Princ wrote:
rishabhmishra wrote:
What is the smallest possible distance between origin and any point on the line \(y=\frac{1}{2}x+50\)?

A. 25
B. \(20\sqrt{5}\)
C. 50
D. \(50\sqrt{2}\)
E. 100


Answer is B
The Smallest distance between the origin and any point on the line \(y=\frac{1}{2}x+50\) would be length of perpendicular drawn
from origin to line \(y=\frac{1}{2}x+50\).

assume the equation of line passing through origin and perpendicular to line \(y=\frac{1}{2}x+50\) be y= mx +c.
as two line are perpendicular and non vertical, product of their slope would be -1.
so m = -2.
as (0,0) lies on y= mx +c, c should be zero.
So equation of perpendicular line to \(y=\frac{1}{2}x+50\) and passing through origin is y=-2x.
Point of intersection of these line would x= -20 and y = 40.

Distance of this point from origin = \(\sqrt{[(-20-0)^2 +(40-0)^2]}\) =20\(\sqrt{5}\)


Two other approach are as follows
1)
The shortest distance between a point to a line =the length of a perpendicular line segment from the line to the point.
The Perpendicular distance from a point \((x_0, y_0)\) to a line \(ax+by+c=0\) is given by the formula:

\(D=\frac{|ax_0+by_0+c|}{\sqrt{a^2+b^2}}\)
putting(0,0) in above formula and equation of line in form of \(-\frac{1}{2}x+y-50=0\) , we get
\(D=\frac{|-50|}{\sqrt{(-1/2)^2+1^2}}\)
\(D=\frac{2*50}{\sqrt{5}}\)
D=\(20\sqrt{5}\)

2) is attached as image
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IMG_20180329_134537.jpg [ 454.42 KiB | Viewed 435 times ]


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What is the smallest possible distance between origin and any point on &nbs [#permalink] 29 Mar 2018, 01:08
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