Shayaan_Mustafa
  • Shayaan_Mustafa
Find the equation for the family of lines tangent to the circle with center at the origin and radius 3.
Mathematics
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SOLVED
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katieb
  • katieb
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Shayaan_Mustafa
  • Shayaan_Mustafa
I know, use the equation y=mx+c. But how?
Hero
  • Hero
First draw the circle
Shayaan_Mustafa
  • Shayaan_Mustafa
|dw:1332964463602:dw|

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Hero
  • Hero
Yes, but, well I also believe we should apply equation of a circle as well to this
Shayaan_Mustafa
  • Shayaan_Mustafa
tangent is perpendicular to the radius of the circle. and slope of perpendicular line are m1=-1/m2. right??
Hero
  • Hero
yes
Shayaan_Mustafa
  • Shayaan_Mustafa
let us simplify the question first. we just find the equation of tangent line and then find the family. OK
Hero
  • Hero
Let me get back to you regarding this question. I think they want more than just the equation of a tangent line at one point on the circle. I think they want an equation of a tangent for any point on the circle
Shayaan_Mustafa
  • Shayaan_Mustafa
yes yes. exactly. family of line implies this idea as you said. but if we find equation for one tangent the we can find equation for any tangent. family too.
Shayaan_Mustafa
  • Shayaan_Mustafa
did you get my idea?
Hero
  • Hero
if you found an equation for one tangent, you would not be able to find it for another tangent. That would only work if (3,0), (-3,0)(0,3)(0,-3) were the points
Hero
  • Hero
You would already have to know the points in advance for any other point on the circle in order to find the equations of tangent lines
Shayaan_Mustafa
  • Shayaan_Mustafa
look Hero. y=-x+c is the family of line which has same slope. do you know this idea? If so then why not for above question?
Hero
  • Hero
What concepts are you studying in class regarding this? Are you working from a textbook? If so, tell me the name of the section you're working on.
Shayaan_Mustafa
  • Shayaan_Mustafa
Calculus by howard anton. 7th edition
Hero
  • Hero
Oh, I see. This is calculus. I thought you were doing high school geometry
Shayaan_Mustafa
  • Shayaan_Mustafa
it is university level.
TuringTest
  • TuringTest
our circle is\[x^2+y^2=9\]the formula for a tangent line at a point 'a' is\[y-y(a)=y'(a)(x-a)\]I think we should differentiate implicitely to get y'
Shayaan_Mustafa
  • Shayaan_Mustafa
no. differentiation is not include up to this section. so we can't differentiate it.
TuringTest
  • TuringTest
eh? um.... then how is it calculus?
Shayaan_Mustafa
  • Shayaan_Mustafa
look look.. it is 1st chapter named functions and differentiation is starting from chap#3
Shayaan_Mustafa
  • Shayaan_Mustafa
no one can help me on this problem :(
TuringTest
  • TuringTest
ok um... let's see if we can be creative based on what we know from functions and trig the following are tangent lines to the circle\[x=\pm3\]\[y=\pm3\]that covers four sides|dw:1332966190330:dw|now we need to develop some formulas for the areas in between the top, bottom, left, and right
TuringTest
  • TuringTest
hm...|dw:1332966361060:dw|here is a tangent line at some mystery point, I wonder if we can divine a formula for the slope of it
Shayaan_Mustafa
  • Shayaan_Mustafa
heheheheh... divine..
TuringTest
  • TuringTest
|dw:1332966453442:dw|maybe we can use some trig here...
Hero
  • Hero
Good luck with that
TuringTest
  • TuringTest
yeah, maybe a different line of reasoning let me start a little earlier
Shayaan_Mustafa
  • Shayaan_Mustafa
OK OK guys. I got my answer. I have solved the mystry. Come here and see me.
TuringTest
  • TuringTest
|dw:1332966712135:dw|the slope of the black line is... gotta go anyway
Hero
  • Hero
Anybody wanna go on vyew?
Shayaan_Mustafa
  • Shayaan_Mustafa
The circle is given by\[x _{o}^{2}+y _{o}^{2}=9\]right?
Hero
  • Hero
okay....
Shayaan_Mustafa
  • Shayaan_Mustafa
as slope of the radius is given by \[y _{o}/x _{o}\]in general right?
Shayaan_Mustafa
  • Shayaan_Mustafa
so the slope of the tangent line which is also perpendicular to the radius of the circle has the slope of \[-x _{o}/y _{o}\], right?
Hero
  • Hero
This is all general knowledge stuff Shayaan. The problem I had was I didn't exactly know the specific approach to get to the solution. But I know all of these elements.
Shayaan_Mustafa
  • Shayaan_Mustafa
equation for the tangent line is given as\[y=mx+c\]put slope of tangent line we get,\[y=(-x _{o}/y _{o})x+c\]
Shayaan_Mustafa
  • Shayaan_Mustafa
\[y _{o}=\pm \sqrt{9-x _{o}^{2}}\]
Shayaan_Mustafa
  • Shayaan_Mustafa
now I am confused.
Shayaan_Mustafa
  • Shayaan_Mustafa
sorry guys now i am unable to solve this stuff.
Hero
  • Hero
Just continue simplifying
Hero
  • Hero
difference of squares
Hero
  • Hero
assuming everything you did is right. Or maybe just leave the answer as is
Shayaan_Mustafa
  • Shayaan_Mustafa
let me solve it again then I post it. thnx for contribution.
anonymous
  • anonymous
is this the question you whant me to answer?
Shayaan_Mustafa
  • Shayaan_Mustafa
yes myko.
anonymous
  • anonymous
k
anonymous
  • anonymous
Use the parametric equation for the circle C = (3cost,3sent) where 0<=t<=2Pi the equation for the tangent unit vector to the circle would be t = (-3sent,3cost)/3sqrt2 so the tangent at the point (a,b) would be (x,y) = (a,b)+pt where p goes from 0 to infinity
Shayaan_Mustafa
  • Shayaan_Mustafa
thnx.
anonymous
  • anonymous
yw
TuringTest
  • TuringTest
Sorry I had something to attend to.... I would perhaps have said something like\[y-y_0=\frac{y_0}{x_0}(x-x_0)=\frac{y_0}{x_0}x-y_0\implies y=\frac{y_0}{x_0}x\]and we have that\[y_0=\pm\sqrt{9-x_0^2}\]so sub that in and we get\[y=\pm\sqrt{(\frac{3}{x_0})^2-1}\]if you want to avoid parametric nonsense not sure if they want you top break it into a step function though, because the \(\pm\) makes it not a function. oh well..
Shayaan_Mustafa
  • Shayaan_Mustafa
hmmm.. this seems familiar much.
TuringTest
  • TuringTest
oh it just says "find the equation" so I would leave it with the \(\pm\)
Shayaan_Mustafa
  • Shayaan_Mustafa
yes..
TuringTest
  • TuringTest
because it does not require the family of lines to be described by a function
TuringTest
  • TuringTest
oh I forgot the x
Shayaan_Mustafa
  • Shayaan_Mustafa
Thnx a lot Turning Test and everyone who helped me on this complex question.
TuringTest
  • TuringTest
\[y=\pm x\sqrt{(\frac{3}{x_0})^2-1}\]
TuringTest
  • TuringTest
you're welcome :)

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