2017-11-04 Sa Oct 消参

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2017-11-04 Sa

$ P(-3, 0) $ 在圆C $ (x-3)^2 + y^2 = 8^2 $ 内，动圆M与圆相切且过P点，求M点轨迹。

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设切点 $ A(a, b) $，圆心 \(M(x, y)\)，则有 \(R_M = MA = MP = R_C-MC\)：

\[ \left\{ \begin{aligned}
(a-3)^2 + b^2 &= 8^2 \\
\sqrt{(x-a)^2+(y-b)^2} &= \sqrt{(x+3)^2+y^2} \\
&= 8-\sqrt{(x-3)^2+y^2}
\end{aligned} \right. \]

Maxima:

solve([(a-3)^2 + b^2 = 8^2, sqrt((x-a)^2+(y-b)^2) = sqrt((x+3)^2+y^2), sqrt((x+3)^2+y^2) = 8-sqrt((x-3)^2+y^2)],[x]);
solve([(a-3)^2 + b^2 = 8^2], [a])

(%i4) solve([(a-3)^2 + b^2 = 8^2], [a]);
                                    2                  2
(%o4)           [a = 3 - sqrt(64 - b ), a = sqrt(64 - b ) + 3]

solve([sqrt((x+3)^2+y^2) = 8-sqrt((x-3)^2+y^2)], [x])

解不出来……

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UPD 2017-11-10 Fr 10:39PM

周一在学校的时候想了一下，由 $ (a-3)^2 + b^2 = 8^2 $ 可知\(a\) \(b\)关系，由 $ \sqrt{(x+3)^2+y2} = 8-\sqrt{(x-3)^2+y2} $ 可知 \(x\) \(y\) 关系，这样就只剩下两个未知数，然后再带入最后一个方程就行了。

I simplfied in hand but the last step involved a polynomial with too much terms. So I decided to go to Maxima at weekend.

Maxima:

Maxima 5.25.0 http://maxima.sourceforge.net
using Lisp Clozure Common Lisp Version 1.7-r14925M  (WindowsX8632)
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) solve([(a-3)^2 + b^2 &= 8^2], [a])

Maxima 5.25.0 http://maxima.sourceforge.net
using Lisp Clozure Common Lisp Version 1.7-r14925M  (WindowsX8632)
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) solve([(a-3)^2 + b^2 = 8^2], [a])
;
                                    2                  2
(%o1)           [a = 3 - sqrt(64 - b ), a = sqrt(64 - b ) + 3]
(%i2) a
;
(%o2)                                  a
(%i3) a;
(%o3)                                  a
(%i4) solve([sqrt((x-a)^2+(y-b)^2) = 8-sqrt((x-3)^2+y^2)], [y])
;
             2            2            2    2              2    2
(%o4) [sqrt(y  - 2 b y + x  - 2 a x + b  + a ) = 8 - sqrt(y  + x  - 6 x + 9)]
(%i5) solve([sqrt((x-a)^2+(y-b)^2) = 8-sqrt((x-3)^2+y^2)], [y]);
             2            2            2    2              2    2
(%o5) [sqrt(y  - 2 b y + x  - 2 a x + b  + a ) = 8 - sqrt(y  + x  - 6 x + 9)]
(%i6) solve([(x-a)^2+(y-b)^2 = 64 + (x-3)^2+y^2 - 16*sqrt((x-3)^2+y^2)], [y])
;
                       2    2                             2    2
              16 sqrt(y  + x  - 6 x + 9) - 2 a x + 6 x + b  + a  - 73
(%o6)    [y = -------------------------------------------------------]
                                        2 b
(%i7)

Seems I don't know how to use Maxima to solve equation correctly...

I gave it the wrong equation in %i4.. 'a' should not be involved. Try again:

Maxima 5.25.0 http://maxima.sourceforge.net
using Lisp Clozure Common Lisp Version 1.7-r14925M  (WindowsX8632)
Distributed under the GNU Public License. See the file COPYING.
Dedicated to the memory of William Schelter.
The function bug_report() provides bug reporting information.
(%i1) solve([sqrt((x+3)^2+y^2) = 8-sqrt((x-3)^2+y^2)], [y]);
                   2    2                        2    2
(%o1)       [sqrt(y  + x  + 6 x + 9) = 8 - sqrt(y  + x  - 6 x + 9)]
(%i2) solve([(x+3)^2+y^2 = 64 + (x-3)^2+y^2 - 16*sqrt((x-3)^2+y^2)], [y]);
Is  3 x - 16  positive, negative, or zero?

negative
;
                                     2                          2
                  sqrt(7) sqrt(16 - x )      sqrt(7) sqrt(16 - x )
(%o2)      [y = - ---------------------, y = ---------------------]
                            4                          4
(%i3)

Well.. let's do it step by step.

\[ \left\{ \begin{aligned}
(a-3)^2 + b^2 &= 8^2 \\
\sqrt{(x-a)^2+(y-b)^2} &= \sqrt{(x+3)^2+y^2} \\
8-\sqrt{(x-3)^2+y^2} &= \sqrt{(x+3)^2+y^2}
\end{aligned} \right. \]