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

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Manager
Joined: 23 Sep 2016
Posts: 233
What is the smallest possible distance between origin and any point on  [#permalink]

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26 Mar 2018, 22:52
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Difficulty:

65% (hard)

Question Stats:

51% (02:05) correct 49% (01:25) wrong based on 76 sessions

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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
Senior Manager
Joined: 22 Feb 2018
Posts: 420
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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

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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Senior Manager
Joined: 22 Feb 2018
Posts: 420
What is the smallest possible distance between origin and any point on  [#permalink]

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29 Mar 2018, 01:08
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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

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
Attachments

IMG_20180329_134537.jpg [ 454.42 KiB | Viewed 793 times ]

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