anonymous
  • anonymous
By finding the points of intersection, show that the line through (0,5) and (3,9) is tangent to the circle x*2+y^2=9
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
you got the equation for the line?
anonymous
  • anonymous
thats all the question give
anonymous
  • anonymous
ok let me rephrase the question can you find the equation for the line?

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anonymous
  • anonymous
|dw:1443226608217:dw|
anonymous
  • anonymous
??
anonymous
  • anonymous
yeah good
anonymous
  • anonymous
so now to find the point of intersection replace \(y\) in \(x^2+y^2=9\) by \(\frac{4}{3}x+5\) and solve \[x^2+(\frac{4}{3}x+5)^2=9\] and see that there is only one solution to that quadratic
anonymous
  • anonymous
x^2+(16/9)x^2+(40/3)x+25=9 (25/9)x^2 + (40/3)x=-16
anonymous
  • anonymous
should get \[(25 x^2)/9+(40 x)/3+25 = 9\]
anonymous
  • anonymous
oh yeah, what you got
anonymous
  • anonymous
multiply both sides by 9
anonymous
  • anonymous
\[25x^2+120x+144x=0\] which , by some miracle , is a perfect square
anonymous
  • anonymous
\[(5x+12)^2=0\\ x=-\frac{12}{5}\] is the only solution
anonymous
  • anonymous
typo there, i meant \[25x^2+120x+144=0\]
anonymous
  • anonymous
ok so thats how we show that the line through the given points is tangent to the circle?
anonymous
  • anonymous
x^2+y^2=9 -12/5+y^2=9 y^2=57/5 y=sq.rt (285)/5
anonymous
  • anonymous
@satellite73
anonymous
  • anonymous
this\[25x^2+120x+144=0\]s the quadratic equation you get
anonymous
  • anonymous
the fact that it has one solution (not 2, not 0) is what tells you that the line is tangent to the graph
anonymous
  • anonymous
ok and if it had 2 solutions then its not tangent to the graph?
anonymous
  • anonymous
yes it would cut the circle twice
anonymous
  • anonymous
and if you had no solutions, it would not cross the circle at all
anonymous
  • anonymous
ok
anonymous
  • anonymous
thank you

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