anonymous
  • anonymous
Find the equation of the tangent to the following parabola 3y^2+16x=0 ; two perpendicular tangents one of the two point of contact being (-3,4)
Mathematics
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SOLVED
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katieb
  • katieb
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anonymous
  • anonymous
Hi!!
anonymous
  • anonymous
@SolomonZelman
SolomonZelman
  • SolomonZelman
Okay, I will reinterpret the question for you.

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SolomonZelman
  • SolomonZelman
A horizontal parabola \(3y^2+16x=0\) will have the reflection of what is above the x-axis, below the x-axis. (Read this sentence carefully) Thus, at x=-3 there are going to be two tangent lines. And you have to choose the tangent line to the curve at (-3,4) and (-3,-4), BUT you have to choose (-3,4).
SolomonZelman
  • SolomonZelman
So our function can be also written as: \(3y^2+16x=0\) \(3y^2=-16x\) \(y^2=-\frac{16}{3}x\) \(y=\color{red}{+} \sqrt{-\frac{16}{3}x}\)
SolomonZelman
  • SolomonZelman
I am choosing only the top half of the functio because the point that we are considering is on the top half of the function.
anonymous
  • anonymous
ok
SolomonZelman
  • SolomonZelman
So, \(f(x)= \sqrt{-\frac{16}{3}x}\) find the equation of the tangent line, at the point (-3,4).
SolomonZelman
  • SolomonZelman
What would be the first thing to do here?
anonymous
  • anonymous
find the slope?
SolomonZelman
  • SolomonZelman
yes, and to do that you have to find the derivative.
SolomonZelman
  • SolomonZelman
What is the derivative of our function ?
anonymous
  • anonymous
y'=1/2*-16/3x^-1/2 y'=(-8/3)x^-1/2 y'=-8/3sq.rtx
SolomonZelman
  • SolomonZelman
correct yourself.
anonymous
  • anonymous
\[y'=\frac{ -8 }{ 3\sqrt{x} }\]
SolomonZelman
  • SolomonZelman
yes, but your quantity inside the square root is?
SolomonZelman
  • SolomonZelman
\(\color{#226633}{f(x)= \dfrac{-8}{3\sqrt{-\dfrac{16}{3}x}}}\)
SolomonZelman
  • SolomonZelman
Now, the slope at x=-3 is f\('\)(-3). Can you find that ?
anonymous
  • anonymous
f(-3)=-2/3
SolomonZelman
  • SolomonZelman
yes, f\('\)(-3)=-2/3
SolomonZelman
  • SolomonZelman
https://www.desmos.com/calculator/9yd8w5xuyf

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