2017-11-04 Sa Oct 消参

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。

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……

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.

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

巴特西

2017-11-04 Sa Oct 消参

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