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liq [111]
3 years ago
7

The endpoints of CD are C(1, -6)and D(7,5). Find the coordinate of the midpoint, M

Mathematics
1 answer:
yuradex [85]3 years ago
6 0

Answer:

(4, - 0.5 )

Step-by-step explanation:

Given endpoints (x₁, y₁ ) and (x₂, y₂ ) then the midpoint is

[ \frac{x_{1}+x_{2}  }{2} , \frac{y_{1}+y_{2}  }{2} ]

Here (x₁, y₁ ) = C(1, - 6) and (x₂, y₂ ) = D(7, 5) , then

midpoint = ( \frac{1+7}{2} , \frac{-6+5}{2} ) = \frac{8}{2} , \frac{-1}{2} ) = (4, - 0.5 )

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At what points does the curve r(t) = ti (4t − t2)k intersect the paraboloid z = x2 y2? (if an answer does not exist, enter dne.)
Aleonysh [2.5K]
\mathbf r(t)=x(t)\,\mathbf i+y(t)\,\mathbf j+z(t)\,\mathbf k=t\,\mathbf i+(4t-t^2)\,\mathbf k
\implies \begin{cases}x(t)=t\\y(t)=0\\z(t)=4t-t^2\end{cases}

So the paraboloid is given by

z(t)=x(t)^2y(t)^2\iff 4t-t^2=0

which holds for t^2-4t=t(t-4)=0, i.e. when t=0 and t=4.

To make sure this is correct, plug in t=0 and t=4 to find out the value of \mathbf r(t), and see if these points are on the paraboloid.

\mathbf r(0)=0\,\mathbf i+0\,\mathbf j+0\,\mathbf k\implies (0,0,0)
0=0^2\times0^2

\mathbf r(4)=4\,\mathbf i+0\,\mathbf j+(4\times4-4^2)\,\mathbf k\implies (4,0,0)
0=4^2\times0^2

Both of these hold, so the answer is correct.
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3 years ago
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Read 2 more answers
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