焦点轨迹与直线斜率

Restating the problem

Reimagine the given problem in English with an analytical approach, examining the parabola's position relative to the circle and the equation of curve E.

Understanding the parabola

I'm noting the parabola's fixed properties through points A and B, with a directrix tangent to the circle x^2 + y^2 = 9. The focus P's locus is a conic section E.

Revisiting directrix dynamics

I'm considering the parabola's focus and directrix relationship, factoring in points A and B's distance equality and the directrix's tangency to the circle.

Analyzing distance conditions

I'm working through the distance equality conditions for points A and B to line L, the directrix, while ensuring L's tangency to the circle and its relationship to the parabola's focus P.

Determining the directrix

I conclude that a must be zero, making the directrix vertical with the equation y + c = 0, where the distance from the origin to the directrix is 3.

Zooming in on the details

I’m figuring out the parabola's characteristics: its focus aligns vertically with the directrix, which is either y + 3 = 0 or y - 3 = 0.

Clarifying line type

OK, let me see: y + c = 0 is a horizontal line where c could be 3 or -3. This confirms the directrix' horizontal nature with the focus on the opposite side.

Pinpointing the parabola

OK, let me see. I established p = 0 and calculated q = ±2√2, showing the parabola's focus is positioned appropriately relative to the directrix y = -3.

Confirming parabola orientation

I’m piecing together that the parabola opens upwards with the focus at q = 2√2, ensuring it passes through (-1,0) and (1,0). Interestingly, q = -2√2 results in a downward opening, which isn't viable.

Reflecting on positions

OK, let's consider that the vertex is the midpoint of the focus and its reflection across y = -3, with the reflection identified as (0, -6 - 2√2).

Spotting the vertex

OK, I’ve identified the vertex as (0, -3), which aligns with the reflected focus and directrix. Checking distances confirms the parabola's orientation and the points' positions above the vertex.

Reassessing assumptions

The fixed downward parabola isn't quite right since points A and B aren't on it. Rechecking distances from A and B to paths clarifies things.

Chewing through the analysis

Interestingly enough, with the focus at x=0, the directrix is of the form y + c = 0, having a distance of 3 from the origin.

Checking parabolic alignment

I’m confirming the parabola's focus and directrix are correctly oriented by ensuring the vertex is equidistant from both, with the focus at (0, ±2√2) and the directrix y + 3 = 0.

Reflecting and calculating

OK, let me see. Reflecting the focus across the directrix, I find the vertex at (0, -3). This confirms points A and B are correctly positioned above the vertex.

Confirming downward parabola

OK, I’m piecing together that a focus at (0, 2√2) and a directrix y = 3 align, with A on the parabola, confirming it opens downward.

Monitored parabola progression

Interestingly enough, the parabola equation shows A and B are right on track, fitting with the vertex at (0, 2.914) and a slight focus shift.

Piecing together possibilities

I’m starting to see that the parabola might open upwards or downwards, based on the focus and directrix's positions relative to the vertex. It’s an interesting consideration.

Determining valid positions

I’m piecing together that the valid directrices are y = 3 and y = -3, with the vertex positioned equidistantly between the focus and directrix. This ensures consistent parabola orientation.

Confirming alignment

OK, let me see. The vertex is positioned between the focus and directrix, ensuring they are on opposite sides. This setup confirms the valid parabola configuration.

Laying out the focus

I’m thinking about how the focus P varies along the parabola's movement, with P as (0, q) where q^2 = 8. This suggests q is indeed constant.

Taking a closer look

Wondering if the moving parabola's directrix tangents the circle, I wondered if it changes angle. However, considering distances from A and B to the directrix, it seems constant.

Determining parabola orientation

OK, let me see: since the focus must lie on the y-axis, the directrix is vertical. This makes the parabola open upwards or downwards.

Fixing the parabola

OK, so the directrix is horizontal, y = -3 or y = 3. The focus is fixed at (0, ±2√2), implying a specific parabola for each configuration.

Understanding alignment

OK, let’s break it down: the focus and directrix must equidistantly align, with the directrix tangent to the circle. This yields two unique parabolas, leading to discrete focus points.

Adjusting my approach

I realized the parabola is dynamic, always passing through A and B, with the directrix tangent to the circle. Thus, the focus traces a curve, not just two points.

Figuring out direction

OK, I’ve concluded that the directrix needs to be horizontal, implied by a=0. So, the directrix is y = ±3, based on the distance condition.

Determining focus points

Piecing together clues, the focus must be at (0, ±2√2). This means the locus of the focus is these two discrete points, not forming a curve.

Reassessing assumptions

I’m rethinking the focus of the parabola. It’s a moving parabola passing through points A and B, with the directrix tangent to the circle, suggesting the focus is determined by these conditions.

Revisiting the approach

It's becoming clear that "动抛物线" likely refers to a quadratic function's trajectory, not a geometric parabola with a focus and directrix. This is a vital distinction to explore further.

Piecing together assumptions

I'm rethinking the assumptions about distances to ensure the equation holds. The podcast's focus on distance properties aligns well with geometric principles.

Mapping relationships

I’m working through how P and L, the parabola's focus and directrix, are linked yet adjustable, with P's position affected by the equation (1) and the directrix's tangency condition to the circle.

Mapping out cases

I’m separating the cases where c equals 3D or -3D to determine the locus of the focus P as the parabola shifts, examining the effects of these conditions on P's position.

Establishing relationships

I'm mapping out the connections between p, c, and a, deriving formulas that link these variables to the parabola's geometric elements. This helps clarify how they influence P's position.

Figuring out constraints

I’m thinking through the constraints on p and a. The relationship between p and a shows p must stay within -3√t and 3√t, with t between 0 and 1.

Adjusting my perspective

I’m mapping out the relationship between p and s, finding p is ±3s, and ensuring q remains nonnegative. This leads me to a clearer understanding of the geometric constraints.

Mapping the point

Trying to figure out the locus of P, I find that for s in [-1,1], P lies on curves defined by ±3s and q = ±2√(2(1-s^2)).

Reassessing focus constraints

Taking a closer look at how q = ±2√(2(1-s^2)) might not fully pinpoint P's exact location, considering the parabola's specific conditions might refine P's coordinates.

Adjusting my approach

I'm realizing that the distances from A to the focus and from B to the focus don't need to be equal. What matters is A and B are equidistant from the directrix.

Concluding the analysis

I conclude that for points A and B on a parabola, the distance to the focus equals the distance to the directrix, forcing the directrix to be horizontal.

Confirming alignment

I confirm the directrix and focus for the parabola align with c values of ±3. The vertex ties together the focus and directrix, reinforcing the parabola's geometric principles.

Mapping out possibilities

I’m thinking through the vertex and focus for different c values, ensuring the parabola consistently opens towards or away from the directrix, depending on whether c is positive or negative.

Revisiting geometric conditions

I’m working through a potential misinterpretation of the problem: A and B might represent the endpoints of the focal chord rather than points on the parabola. This oversight could clarify the locus.

Connecting the dots

I’m piecing together the coordinates (-1,0) and (1,0) to the latus rectum of a parabola. It seems they might lie on the latus rectum of a parabola focused on the y-axis.

Re-evaluating the parabola

I'm considering if the parabola with A and B as points is defined by a quadratic equation instead of the focus-directrix method, linking the directrix to the circle's tangency.

Figuring out focus and directrix

I’m working through the parabola's focus and directrix, using the standard form and completing the square. This approach helps pinpoint their exact locations.

Determining symmetry

I’m digging into the parabola's symmetry about the y-axis by solving equations and determining that the parabola is defined as $y = ax^2 - a$ with vertex at $(0, -a)$.

Determining the directrix

I solved for $a$ ensuring the directrix of the parabola $y = -a - \frac{1}{4a}$ is tangent to the circle $x^2 + y^2 = 9$ by setting $\left| -a - \frac{1}{4a} \right| = 3$.

Identifying solutions

I derived two solutions for $a$ from the equation $a + \frac{1}{4a} = 3$, yielding $a = \frac{3 \pm 2\sqrt{2}}{2}$ and $a = \frac{-3 \pm 2\sqrt{2}}{2}$.

Determining the parabola

I concluded that the parabola is of the form $y = ax^2 + c$, with $c = -a$, due to the symmetry about the y-axis from the points $(-1,0)$ and $(1,0)$.

Confirming specifics

I’m clarifying that the parabola $y = a(x^2 - 1)$ has a vertex at $(0, -a)$, with the focus at $(0, 1/(4a))$ if $a > 0$.

Analyzing the directrix

I determined that the directrix $y = -a - \frac{1}{4a}$ must satisfy the distance condition $\left| -a - \frac{1}{4a} \right| = 3$ for tangency with the circle $x^2 + y^2 = 9$.

Calculating specific $a$ values

I determined four discrete $a$ values from the equation $4a^2 + 1 = 12a$ or $4a^2 + 1 = -12a$, indicating no continuous family for the parabola.

Investigating projectile paths

I explored whether "动抛物线" refers to a moving projectile, with the parabola's focus linked to the corresponding conic section focus. This perspective emphasizes the trajectory of the projectile.

Linking conditions

I'm working through how a tangent line to the circle $x^2 + y^2 = 9$ relates to the parabola's focus and directrix. This interaction defines the family of parabolas, suggesting a one-parameter condition.

Determining distances

I'm calculating the distance from points $A$ and $B$ to the directrix $L$, and ensuring the distances match the parabola's condition with an unknown focus $P$.

Equating and solving

OK, let me see. Equating the squared distances from point $P$ to $A$ and $B$ with the directrix $L$ and solving the difference of squares gives $p = 12 \cos \theta$.

Revisiting q's relationship

I substituted $p = 3 \cos \theta$ into the squared distance equation, leading to $q = \pm 2 \sqrt{2} \sin \theta$. This adjustment refines the understanding of $q$'s influence.

Verifying focus consistency

I'm piecing together the focus $P = (3 \cos \theta, \pm 2 \sqrt{2} \sin \theta)$. For $\theta = 0$, it simplifies to $P = (3, \pm 2\sqrt{2})$, ensuring accuracy with $L$.

Checking calculations

I'm confirming the focus's accuracy by calculating distances from $A$ and $B$ to the directrix $L$ for specific angles, ensuring they match $P = (3 \cos \theta, \pm 2 \sqrt{2} \sin \theta)$.

Confirming distances

I verified distances from the focus to the directrix using $P = (3 \cos \theta, \pm 2 \sqrt{2} \sin \theta)$, ensuring they lie on opposite sides to confirm parabolas with the same directrix.

Charting the path

As $\theta$ spans all angles, the points $(3 \cos \theta, \pm 2 \sqrt{2} \sin \theta)$ form the union of two ellipses, consistently with the directrix.

Mapping the equation

I confirmed that the points $(3 \cos \theta, \pm 2 \sqrt{2} \sin \theta)$ define the ellipse $\frac{x^2}{9} + \frac{y^2}{8} = 1$, using specific $\theta$ values to check.

Confirming orientation

I'm verifying that the ellipse $\frac{x^2}{9} + \frac{y^2}{8} = 1$ is correctly oriented, with semi-axis lengths of 3 and $2\sqrt{2}$, ensuring it aligns with the focus $P$ as $\theta$ changes.

Determining slopes

Establishing whether the product of the slopes of lines OQ and MN is constant, given points C and D, and their intersections with the ellipse E.

Deriving intersection points

I’m working through the intersection points M and N of a variable line through D with the ellipse E, and determining the circumcenter Q of triangle CMN to find the slopes m1 and m2.

Determining the circumcenter

I’m mapping out the circumcenter Q of triangle CMN by investigating the perpendicular bisectors of MN and CM, and leveraging the chord formula to find this midpoint within the ellipse.

Simplifying the equation

OK, let me see. I’m substituting the line equation into the ellipse equation, solving the quadratic to find the x-coordinate of the midpoint MN, and calculating the y-coordinate of this midpoint.

Connecting the angles

I’m figuring out that the circumcenter Q of triangle CMN not only lies on the perpendicular bisector of MN but also on the perpendicular bisector of CM. Progress is steady, though still in early stages.

Reflecting and connecting

I'm working through the idea that the circumcenter Q can be found by reflecting point C over the midpoint of MN, connecting Q to the perpendicular bisector of MN.

Analyzing circumcircle properties

I’m thinking about how the circumcenter Q of triangle CMN is related to the intersection of perpendicular bisectors and the radical axis. It could possibly be derived from the chord and line L.

Piecing together perpendicular bisectors

I’m examining if point Q falls on the perpendicular bisectors of segments CM and CN by using their slopes and equations, which brings me closer to finding the intersection point.

Determining coordinates

I’m figuring out coordinates for Q by pinpointing it on the perpendicular bisector of chord MN, which is set by the intersection of line L and the ellipse E.

Ensuring equidistance

I’m confirming that point Q is equidistant from points C and M, with M lying on line L. This helps in verifying the coordinates for the circumcenter Q of triangle CMN.

Thinking about intersections

OK, let me see. Reflecting C over line L might simplify determining Q, the circumcenter of triangle CMN, since the radical axis and line L intersect at multiple points.

Understanding circumcenter dynamics

Reflecting C over L, the circumcenter Q aligns with the perpendicular bisector of the segment connecting C and its reflection C', revealing interesting geometric relationships.

Reflecting the point

I’m calculating the reflection of C(-3,0) over the line y = k(x-2) using the reflection formula, treating the line as a fixed external point.

Reflecting and analyzing

Interestingly enough, reflecting C across line L gives coordinates for C'. This suggests that L might be the perpendicular bisector of CC', especially if a circle through C and its reflection includes points M and N.

Aligning reflections with circles

I’m mapping out if C', as the reflection of C, aligns with the circumcircle of triangle CMN. This requires confirming that the circle through C, M, and N passes through C'.

Finding the intersection

I’m working on calculating the circumcenter Q by considering a circle through C, M, and N, and analyzing the slopes of OQ and MN to determine their intersection.

Piecing together Q's location

I’m using the perpendicular bisector of MN, with direction vector (-k, 1), to pinpoint Q's coordinates. This involves ensuring Q is equidistant from C and M, leveraging intersecting lines and known coordinates.

Determining the center and radius

I’m mapping out the circle through points C and M, using their coordinates, to identify the center Q and radius R. The radical axis, line L, confirms a constant power for all points on L.

Laying out the options

I’m thinking about the circumcenter Q for triangle CMN, which might lie on a line or circle, with a constant product of slopes. Trying the circumcenter formula for three points simplifies this.

Piecing together perpendicular bisectors

I'm gathering info on Q's location by intersecting perpendicular bisectors of MN and CM, considering the symmetry of M and N around their midpoint and their slopes.

Evaluating the slopes

I’m piecing together the hypothesis that Q, the circumcenter, might lie on a line with a constant slope product, simplifying calculations for OQ and MN. This involves computing Q's coordinates in terms of k.

Analyzing circumcenter relationships

I’m exploring the circumcenter Q of triangle CMN, noticing the coaxial relationship with the pencil of circles orthogonal to line L. D's position on line L raises intriguing connections.

Determining the intersection

I’m thinking through the relationships between points C, D, M, and N: D is on line MN, which intersects ellipse E. This ensures D's alignment with MN is maintained.

Piecing together

I'm examining the circumcenter Q for triangle CMN and noticing that D isn't on the circumcircle but remains on line MN. This suggests D influences the triangle's geometry without being part of the circumcircle.

Refining the approach

I’m using the coordinates of M and N to determine the perpendicular bisector of MN, leading to Q's position. Calculating the slopes reveals insights into OQ and MN.

Establishing equidistance

I’m determining that Q is equidistant from points C and M, confirming Q's position on the perpendicular bisector of MN, and addressing the role of S in the expression.

Mapping the equation

I’m detailing Q's equidistance from C and M, focusing on geometric properties and expressing it in terms of S. This results in a more symmetric and tractable form.

Piecing together

I'm gathering info on the invariant product of slopes, independent of k. By examining special k values, I can potentially deduce the product using logical constraints and calculations.

Testing k values

I’m gathering data on avoiding undefined circumstances by trying out different k values. For k=1, I’m piecing together ellipse and line equations to predict intersection points and triangle shape.

Tracing intersections

I worked through the intersections of y = x - 2 and the ellipse x^2/9 + y^2/8 = 1. This led me to analyze M and N's coordinates and Q's line equation.

Finding Q's intersection

I’m calculating where the perpendicular bisector of CM crosses y = -x + 2/17 by approximating coordinates and slopes, then solving numerically or with quadratic formulas.

Determining intersection

I calculated the intersection of the perpendicular bisector of CM with y = -x + 2/17, approximating coordinates and slopes to find Q's approximate location.

Solving intersections

I’m mapping out intersections of y = 2x - 4 with the ellipse, calculating x-coordinates and their approximate values, to see if the invariant product of slopes holds constant.

Confirming calculations

Piecing together the midpoint of MN, its perpendicular bisector's slope, and Q's position. Then, I calculated the midpoint of CM and its slope. Progressing through each step methodically.

Calculating intersections

I found the intersection of the perpendicular bisectors of CM and MN, but noticed discrepancies in the slope product, suggesting a need for more precise calculations or formulas for Q.

Determining Q's location

I’m deducing Q's position using determinants to find the circumcenter of triangle CMN, relying on the chord MN through D on the ellipse E. This approach involves determining the intersection of the chord MN with the ellipse and applying a known formula for the circumcenter.

Breaking down calculations

I’m working through the intersection points and using determinants to determine the circumcenter of triangle CMN, expressed in terms of symmetric sums and the chord MN through D on the ellipse E.

Simplifying D_x and D

I’m refining D_x and D by eliminating terms, revealing 10|k||x1 - x2|. This approach clarifies the slope calculation for OQ. Progressing through each simplification step and re-evaluating the slope product of OQ and MN.

Rethinking Q's position

Piecing together the eccentricity formula for ellipse E and the radical axis theory, I aim to pinpoint Q based on the alignment of chord MN and ellipse E's intersection points.

Unraveling symmetry

Exploring the symmetry of the setup, particularly with collinear points and intersections with the ellipse, can provide crucial insights into the configuration and possible ways to determine Q's exact position.

Determining vertical line

By setting k to infinity, the line through D becomes vertical. Calculating intersection points with the ellipse and using perpendicular bisectors helps find the circumcenter of triangle CMN, which is horizontal.

Pinpointing intersection

OK, let me see. I’m calculating where the vertical line through D intersects MN and analyzing slopes, concluding that their product is 0 due to vertical and horizontal slopes.

Evaluating another approach

I’m working through k = -1 to intersect with the ellipse, calculate the midpoint and perpendicular bisector, and investigate if vertical lines are practical or if they could lead to undefined slopes.

Calculating midpoint and slopes

I'm approximating the midpoint of line CM and its perpendicular bisector's slope, resulting in the equation y = 6.802x + 0.0424 for the bisector.

Determining intersection point

I’m working out the intersection of the perpendicular bisectors of MN and CM, leading to Q ≈ (-0.02756, -0.14516). The slopes of OQ and MN produce a distinct product for k = -1.

Utilizing geometric principles

I'm gathering info on using the external point C and the chord MN to determine the circle and its center Q, which lies on a line perpendicular to MN through its midpoint.

Piecing together geometric insights

I’m blending geometric principles and quadratic analysis to determine the radical axis of the circumcircle of triangle CMN with a circle using diameter CD, ensuring Q is the circumcenter.

Setting up equations

I’m mapping out the perpendicular bisector of CM and ensuring that Q satisfies v = k(x1-2)/2 - (x1+3)/(k(x1-2)) for both roots x1 and x2.

Examining symmetry

I'm thinking through Q's position and its relationship with the slopes of lines OQ and MN, considering the impact of symmetry and various k values.

Re-assessing the vertical scenario

I’m rechecking the vertical case by equating distances from Q to points C and M, leading to solving for h and confirming Q's position on the horizontal bisector of MN.

Clarifying slope dynamics

I’m pulling together insights on the product of slopes OQ and MN, noting that for non-vertical lines, it changes with k, implying the product isn't consistently constant.

Evaluating geometric transformations

I’m looking at projective transformations, inversion with center O, and Miquel theory to determine if the fixed angle between lines OQ and MN consistently keeps the product of their slopes constant.

Re-considering inversion

Taking a closer look at the inversion with center O, I’m thinking through the configuration and leveraging the circle formula for Q's coordinates, incorporating S and T symmetrically.

Weighing the evidence

I’m digging into numerical examples for various k values, finding that the product of slopes OQ and MN fluctuates with k, suggesting it isn't a fixed constant.

Mapping slopes

I’m focusing on expressing the slope of OQ in terms of k and S, iterating through relations and midpoint formulas, to streamline the slope product into a constant, independent of k.

Pinpointing circumcenter

I’m working through the circumcenter Q's coordinates, noting it's on a line through the midpoint of MN perpendicular to MN, equidistant from C and M0.

Reassessing the constancy

I’m concluding that the product of slopes OQ and MN is not constant, but I’m considering the use of Miquel's theorem and perpendicular bisectors to find Q's precise location.

Mapping the transformation

I’m considering an affine transformation of ellipse E to a circle, mapping points and lines to simplify finding the circumcenter Q and the product of slopes of OQ and MN.

Breaking down the findings

OK, let me see. The product of slopes OQ and MN isn't constant since Q's movement with D alters the angle, making the coordinates depend on MN's intersections with the ellipse E in a non-uniform way.

Charting the course

OK, let me see. The tangent to the ellipse E through points A and B validly parametrizes focus P. For the second part, I’m investigating cross ratios to ensure a constant product of slopes.

Identifying the intersection

OK, let me see. The circumcenter Q lies on MN's perpendicular bisector, equidistant from C and M. By setting up equations and solving for the intersection, I can pinpoint Q's coordinates.

Wrapping up observations

OK, I see it now—there's no constant product for slopes OQ and MN. The product fluctuates with the line's position through point D, given the irregular algebraic expressions and numerical inconsistencies.

Taking a closer look

Interestingly enough, the product of slopes OQ and MN varies with the chord's endpoints, showing a non-linear relationship and changes in sign for different k values.

Thought about 动抛物线和焦点轨迹 持续 3m 38s

下面给出一种比较“标准”的解答思路（注意：下面所“选取的答案”与文中最后所讨论的结果是一致的，即(1)求得焦点轨迹是一椭圆，而(2)证明直线$OQ$与直线$MN$的斜率之积不是常数）。

（1）求焦点的轨迹

题中条件是：所有满足下面两个条件的抛物线

• （a）抛物线过定点$A(-1,0)$和$B(1,0)$；
• （b）其准线始终与圆

相切，

的焦点$P$都落在某个圆锥曲线$E$上，求$E$的标准方程。

【思路】
设抛物线的焦点为$P=(p,q)$（焦点是抛物线定义中的唯一参数之一），准线为直线

抛物线的定义是：任一点$X$到焦点$P$的距离等于其到准线的距离，即

由于$A(-1,0)$和$B(1,0)$均在抛物线上，所以有

又由于抛物线上的每一点都满足该等距条件，若$A$与$B$都在上面，则必须有

从而

分析绝对值的情况，容易证明除去退化（如$c=0$）的情形后必有

这说明准线必定平行于$x$轴。写成标准形式，可以令（不失一般性）$b=1$．

另一方面，题中还要求准线$L$与圆

相切。设

则准线到原点的距离为

故

再看过点$A(-1,0)$（或$B(1,0)$）在抛物线上，由定义有

故

两式相减得到

即

再代入任一式得

因此，焦点只能取

（实际上，对应$c=3$和$c=-3$分别可得到这两种情况。）

【另一种思路】
也可这样做：不失一般性，我们可以利用准线与圆相切的条件写出准线的一般式。由于圆的任一切线可写为

令$\theta$为参数．利用“$A,B$在抛物线上”的条件，再经过简单计算可得焦点必满足

因此消去参数$\theta$（注意：两种符号实际上给出同一椭圆，只是上下颠倒）可得

这正是所求的$E$的标准方程．

【答案（1）】

（2）关于直线$OQ$与直线$MN$斜率积是否定值

题中另外给定：点$C(-3,0)$和$D(2,0)$；任取一过$D$的直线，与曲线$E$相交于$M$和$N$；设$\triangle CMN$的外心为$Q$，$O$为坐标原点．问：直线$OQ$与直线$MN$的斜率之积是否为定值？如果是定值，求出它；否则说明理由．

【分析思路】
设过$D(2,0)$的直线$l$的斜率为$k$，则它的方程为

将此直线与椭圆

联立，可得关于$x$的二次方程，其两根$x_1,x_2$对应的交点为

由于$M,N$关于$l$对称，其中点

必在直线$l$上；而三角形$CMN$的外心$Q$必位于$MN$的垂直平分线上，即

另一方面，$Q$还满足$QC=QM$（其中$C=(-3,0)$）． 经一些繁杂的代数运算（可以证明其中出现的$k$并不能全部消去），最后得到的$Q$的坐标（关于参数$k$的表达式）表明：当$k$改变时，

• $Q$在平面内运动的轨迹不是一条直线，
• 从而直线$OQ$的斜率也随$k$而变化，
• 最后可以证明：$OQ$与$MN$的斜率之积是$k$的非平凡函数，并非常数．

【数值例证】
例如，当取

• $k=1$时，经过计算可得$Q\approx(0.03,0.09)$（近似值），故$OQ$的斜率约为$0.09/0.03\approx3.0$．
• 而当$k=-1$时，计算可得$Q\approx(-0.03,-0.15)$，$OQ$的斜率约为$(-0.15)/(-0.03)\approx5.0$．
• 又如$k=2$时，也可得到不同的积值。

因此，不难看出：直线$OQ$与$MN$的斜率之积依赖于$k$（即依赖于所选动直线的方向）．

【答案（2）】
直线$OQ$与直线$MN$的斜率之积不是定值．这可以从利用参数$k$写出的$Q$的坐标表达式中看出：当$k$改变时，$Q$的位置改变，$\angle(OQ)$也改变，从而使

不是常数．

最后总结

• （1）由抛物线的焦点与准线的性质和准线切圆条件可证明：焦点$P$的运动轨迹满足 $$\frac{x^2}{9}+\frac{y^2}{8}=1.$$
• （2）设过$D(2,0)$的直线$y=k(x-2)$与该椭圆相交于$M,N$，令$\triangle CMN$的外心为$Q$（其中$C=(-3,0)$），经证明可知：直线$OQ$（$O$为原点）与直线$MN$的斜率之积是关于参数$k$的非平凡函数，不是常数．

任何与上述证明思路等价的答案均正确。