anonymous
  • anonymous
The slope of the tangent to a curve at any point (x, y) on the curve is -x/y. Find the equation of the curve if the point (2, −2) is on the curve.
Mathematics
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anonymous
  • anonymous
The slope of the tangent to a curve at any point (x, y) on the curve is -x/y. Find the equation of the curve if the point (2, −2) is on the curve.
Mathematics
schrodinger
  • schrodinger
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anonymous
  • anonymous
not sure how to do this one
ganeshie8
  • ganeshie8
what do you know about the relation between `slope of tangent line` and `first derivative` ?
anonymous
  • anonymous
well the first derivative represents the slope of the tangent line on any point of the function you derive

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ganeshie8
  • ganeshie8
You're given that the `slope of tangent line` equals -x/y, so setup an equation using that info
anonymous
  • anonymous
i thought that -x/y was the derivative
ganeshie8
  • ganeshie8
Yes \[\dfrac{dy}{dx}=\dfrac{-x}{y}\]
anonymous
  • anonymous
what do i solve for though ?
ganeshie8
  • ganeshie8
thats the equation ^
ganeshie8
  • ganeshie8
you need to solve the curve \(y\)
anonymous
  • anonymous
so implicit differentiation ?
ganeshie8
  • ganeshie8
An usual algebraic equation involves isolating a "variable", but a differential equation involves solving for a "curve"
ganeshie8
  • ganeshie8
familiar with variable separation ?
anonymous
  • anonymous
yep, give me a sec
ganeshie8
  • ganeshie8
https://www.khanacademy.org/math/differential-equations/first-order-differential-equations/separable-equations/v/separable-differential-equations
anonymous
  • anonymous
okay im getting \[\sqrt{C - x^2}\]
anonymous
  • anonymous
is that right?
ganeshie8
  • ganeshie8
looks partially correct, show me ur work
anonymous
  • anonymous
|dw:1439277292623:dw|
anonymous
  • anonymous
this drawing tool is dreadful
anonymous
  • anonymous
then i isolated y
ganeshie8
  • ganeshie8
leave it as \[y^2=-x^2+C\]
anonymous
  • anonymous
|dw:1439277527370:dw|
ganeshie8
  • ganeshie8
Now plugin the initial condition, \((2, -2)\), and solve \(C\)
ganeshie8
  • ganeshie8
|dw:1439277677058:dw|
anonymous
  • anonymous
|dw:1439277671100:dw|
ganeshie8
  • ganeshie8
|dw:1439277771874:dw|
anonymous
  • anonymous
oh woops , my mind was slipping
anonymous
  • anonymous
okay so know we just plug that into |dw:1439277880677:dw| right?
anonymous
  • anonymous
these area my answer choices by the way x + y = 0 x2 − y2 = −2 x2 + y2 = 16 x2 + y2 = 8 i assume its D
ganeshie8
  • ganeshie8
Yes
anonymous
  • anonymous
thanks for the help
ganeshie8
  • ganeshie8
\[y^2=-x^2+C\] plugin \((2, -2)\) \[(-2)^2=-2^2+C\implies C = 8\] so the solution is \[y^2=-x^2+8\] which is same as \[x^2+y^2=8\]

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