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kenneyfamily
Group Title
Find the distance between P(2, 1) and the line with equation x  2y + 4 = 0.
a. (4√5)/5
b. 0
c. 5/4
d. (4√5)/5
 2 years ago
 2 years ago
kenneyfamily Group Title
Find the distance between P(2, 1) and the line with equation x  2y + 4 = 0. a. (4√5)/5 b. 0 c. 5/4 d. (4√5)/5
 2 years ago
 2 years ago

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wio Group TitleBest ResponseYou've already chosen the best response.2
I can't believe they're making this a multiple choice question. What a wingspan move.
 2 years ago

kenneyfamily Group TitleBest ResponseYou've already chosen the best response.0
can you help me?
 2 years ago

wio Group TitleBest ResponseYou've already chosen the best response.2
Okay, well the distance from it is given my distance formula:\[ d = \sqrt{(x_2x_1)^2+(y_2y_1)^2} \]
 2 years ago

wio Group TitleBest ResponseYou've already chosen the best response.2
In this case those... \(x_1=2\), \(x_2=x\), \(y_1=1\), \(y_2=y\).
 2 years ago

wio Group TitleBest ResponseYou've already chosen the best response.2
I would do \(x_2=x=2y4\) so that you have it all in one variable.
 2 years ago

wio Group TitleBest ResponseYou've already chosen the best response.2
So we get \[ d(y) = \sqrt{((2y4)(2))^2+((y)(1))^2}=\sqrt{(2y2)^2+(y1)^2} \]
 2 years ago

wio Group TitleBest ResponseYou've already chosen the best response.2
That needs to be simplified a bit.
 2 years ago

wio Group TitleBest ResponseYou've already chosen the best response.2
\[ \sqrt{4y^28y+4+y^22y+1}=\sqrt{5y^210y+5} \]
 2 years ago

wio Group TitleBest ResponseYou've already chosen the best response.2
\[ 5y^25y5y+5 = 5y(y1)5(y1)=(5y5)(y1)=5(y1)(y1) \]Nice how these problems seem to work out for us...
 2 years ago

wio Group TitleBest ResponseYou've already chosen the best response.2
\[ d(y)=\sqrt{5(y1)^2}=\sqrt{5}(y1) \]Now we want to minimize this distance function....
 2 years ago

kenneyfamily Group TitleBest ResponseYou've already chosen the best response.0
ok how do we do that?
 2 years ago

wio Group TitleBest ResponseYou've already chosen the best response.2
We take the derivative and find some critical numbers!!
 2 years ago

wio Group TitleBest ResponseYou've already chosen the best response.2
\[d'(y)=\sqrt{5} \]Wait what...
 2 years ago

wio Group TitleBest ResponseYou've already chosen the best response.2
So there are no critical points... which means that there is no minimum.... except for the fact that the distance can't be negative.... So the minimum distance must be 0.
 2 years ago

kenneyfamily Group TitleBest ResponseYou've already chosen the best response.0
so the answer is 0?
 2 years ago

wio Group TitleBest ResponseYou've already chosen the best response.2
We can check by putting the point into the line.\[ (2)+2(1)+4=0\implies 22+4=0\implies 4+4=0\implies 0=0 \]So the point is actually on the line! So the distance is 0!
 2 years ago

kenneyfamily Group TitleBest ResponseYou've already chosen the best response.0
awesome thanks!
 2 years ago
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