Find the orthogonal trajectories of the family of curves. (Use C for any needed constant.)
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Answer:
The interpolating polynomial is .
Step-by-step explanation:
We want to find a quadratic polynomial such that
,
and
. In order to do this let us write
.
Now, evaluating the polynomial in the points -1, 1 and 2 we get
This relations give us a linear system of equations:
where the ,
and
are the unknowns.
The augmented matrix of the system is
In this matrix it is easy to eliminate the 1's of the first column and get
From this matrix we can find the values of each unknown. Notice that the second row gives us that yields
.
Then, the third row means that gives
. So,
.
Finally, the first row is and substituting is
that yields
.
Therefore, the interpolating polynomial is
.