抛物线焦点轨迹

Mapping out the locus

I’m given that the parabola intersects with the circle and its directrix is tangent to the circle. Then, I'm tasked with finding a conic $E$ through point $P$, and analyzing the relationship between slopes of $OQ$ and $MN$.

Analyzing the parabola

I’m setting up equations to represent the parabola, pinpointing its focus and directrix, and ensuring it meets the given intersection constraints with the circle.

Pinpointing tangency

I'm figuring out the conditions for the directrix to touch the circle, ensuring distances from points $A$ and $B$ to the focus and directrix align.

Piecing together

I’m working through the focus and directrix, ensuring their distances and tangency conditions are systematically analyzed to depict the parabola accurately.

Reflecting on geometric constraints

Piecing together how the directrix, tangent to the circle, influences the parabola's geometry, with its distance from the origin defining the reflective tangent line.

Establishing tangent specifics

Interestingly enough, the parabola's geometry hinges on the directrix being 3 units from the origin, leading to a refined tangent line equation. This insight is crucial for further analysis.

Mapping the problem

I'm laying out the conditions for $A$ and $B$ to be on the parabola, using distances from the focus and directrix, and ensuring the tangent condition to the circle is met.

Mapping the parabola

For each $\theta$, I’m identifying the unique directrix and focus, ensuring the parabola intersects at $A$ and $B$, and utilizing known geometric properties to define the vertex and axis.

Aligning axes

The axis of the parabola, perpendicular to the directrix, intersects the origin. By representing the focus in polar coordinates, this configuration is captured accurately.

Mapping the focus

I’m examining the distance from the origin to the directrix and the focus to the directrix, ensuring the tangency condition is satisfied with $|c| = 3$ and $|r - 3|$.

Verifying conditions

Checking distances to ensure the parabola passes through $A$ and $B$, mapping out the relationship between the focus and directrix, and confirming tangency to the circle.

Ensuring alignment

I’m piecing together the distance from $A$ to the directrix, ensuring it aligns with the tangency condition while preserving the signs for consistency.

Confirming distances

OK, let’s make sure the parabola’s distances from points $A$ and $B$ to the directrix align with the tangency condition. This confirms the setup is accurate and consistent.

Charting the axis

I’m mapping out how the parabola's axis impacts the focus and directrix. Since the directrix is tangent to the circle, the possible positions for the focus are on the x-axis.

Analyzing possibilities

I'm considering the focus being on the x-axis, either at $x = 3$ or $x = -3$, and verifying if these align with the parabola's conditions.

Mapping the axis

I'm narrowing down the focus' position on the y-axis, with the directrix at $y = 3$ or $y = -3$, and calculating the distances from $A$ and $B$ to verify alignment.

Pinpointing the axis

Taking a closer look, the parabola's axis is vertical through the directrix $y = 3$ or $y = -3$, with potential focuses at $(0, 2\sqrt{2})$ or $(0, -2\sqrt{2})$.

Verifying the alignment

I'm checking if $A$ and $B$ align with the parabola's focus and directrix, specifically the tangency of the directrix with the circle, using $y = 3$ and $y = -3$.

Tracing the relationship

I’m digging into how rotating the directrix around the circle yields a continuous family of parabolas, with a unique focus for each, ensuring $A$ and $B$ remain on the parabola.

Navigating geometric reflections

Considering how reflecting a circle across the directrix might create a conic, but acknowledging the complexity of applying this standard property to our specific parabola scenario. Progressing through these concepts can be quite intricate.

Laying out the conditions

I'm setting up equations for the parabola, factoring in the distance constraints and the tangency condition to the circle, to identify the focus' position as the directrix rotates.

Mapping out possibilities

I'm exploring different conic shapes for the locus $E$ and noting a reflective property for a point around a tangent circle as potential leads to identify a simpler solution.

Analyzing the hyperbola

I’m thinking about the possibility that $E$ is a rectangular hyperbola, examining the reflection of $P$ across the directrix and using the midpoint of $P$ and $P'$ to determine the vertex.

Understanding geometric reflections

Reflecting the circle across the tangent directrix $l$ reveals a fascinating pattern. The circle's reflection becomes its envelope, and through symmetry, likewise passes through $P$.

Maintaining radius and tangency

It's interesting that reflecting a circle tangent to a line keeps the same radius. The circle's reflection, tangent at the same point, has a center as the origin's reflection.

Reflecting ideas

Considering how reflecting the origin across the tangent directrix $l$ forms a circle with radius 3, centered at this reflection, and possibly including point $P$ as well. This insight helps clarify the geometric setup.

Exploring geometric insights

I’m mapping out how reflecting the origin across the tangent directrix $l$ forms a circle with radius 3, centered at this reflection, and possibly including point $P$ too.

Reflecting the concept

Reflecting the origin across the tangent directrix $l$ creates a circle with radius 3, centered at this reflection, and likely containing point $P$ as well.

Reflecting and tangency

Reflecting the origin across the tangent directrix $l$ creates a circle with radius 3, centered at this reflection, and likely containing point $P$. The center of this circle will be $O'$.

Evaluating geometric shapes

I’m piecing together that the locus $E$ could be an ellipse or hyperbola with foci at $A$ and $B$. Considering pythagoras, the focus might lie on this curve.

Evaluating geometric properties

OK, let me see. The directrix tangent to a circle and the possible focus positions hint that the locus $E$ could indeed be an ellipse or a hyperbola with foci at $A$ and $B$.

Spotting points

Interestingly, reflecting point $P$ across the directrix $y = 3$ creates a circle with radius 3 centered at $O'$, likely containing point $P$.

Reflecting and analyzing

Reflecting point $P$ across the directrix $y = 3$ forms a circle with radius 3 centered at $O'$, likely includes point $P$, and might reveal symmetry with respect to the midpoint of $A$ and $B$.

Tracing patterns

Reflecting on how the parabola's symmetry might imply that the axis equally divides segment $AB$ and confirms $P$ could be on the $y$-axis.

Crafting the equation

Formulating the parabola involves determining the focus and directrix to ensure $A$ and $B$ are on it, given the tangency condition and circle's radius.

Pondering symmetry

I'm pulling together the idea that symmetry around the y-axis might simplify equations by using absolute value properties, especially considering $A$ and $B$ are equidistant from the directrix.

Simplifying angles

I’m working through angles, using cosine and sine identities to express the parabola's radius $r$ in terms of the angle $\theta$, which ultimately simplifies to $r = 3$.

Confirming symmetry

Exploring how $\theta = 0$ and $\theta = \pi$ give the valid directrixes for the parabola, ensuring they pass through both points $A$ and $B$.

Validating points

OK, after rechecking, it’s clear that $\theta = 0$ and $\theta = \pi$ work, pinpointing the focus at $(3,0)$ and $(-3,0)$, both on the circle.

Aligning coordinates

OK, let me see. I’m aligning $cos\theta$ and $r$ to ensure the focus falls on the circle, noting discrete points like $(3,0)$ and $(\pm 2\sqrt{2})$.

Adjusting my approach

I'm examining alternative sign combinations to determine if the locus $E$ represents a continuous conic, or just discrete solutions like points $(3,0)$ and $(\pm 2\sqrt{2})$.

Evaluating sign scenarios

I'm looking at four sign combinations to pinpoint the relationship between $r$ and $cos\theta$, aiming to find a continuous family of solutions or discrete points for the locus $E$.

Assessing conic options

I'm weighing if the locus $E$ is an ellipse or a rectangular hyperbola, given geometric properties like constant sum of distances from two points or constant product of distances to a tangent line.

Challenging surface choice

I'm thinking about whether $E$ is a rectangular hyperbola with asymptotes along the coordinate axes, testing solutions to see if partial solutions fit this conic model.

Pinpointing geometric shape

I'm establishing that the standard equation of $E$ is an ellipse: $\frac{x^2}{9} + \frac{y^2}{8} = 1$, ensuring it intersects the given points.

Piecing it together

I'm beginning to see that the locus $E$ is an ellipse $\frac{x^2}{9} + \frac{y^2}{8} = 1$, intersecting the given points, and suspecting the slopes of $OQ$ and $MN$ are constant.

Tracing lines and slopes

I'm working through the problem by determining the circumcenter of triangle $CMN$, identifying altitudes, and connecting $O$ and $Q$ to tackle the problem.

Piecing together slopes

I’m analyzing the slopes of lines $OQ$ and $MN$ to find if their product is constant, leveraging the centers of their perpendicular bisectors for a systematic approach.

Piecing together

OK, so C is on the ellipse $E$. Considering a chord through $D$ intersecting $E$ at $M$ and $N$, I’m starting to see how this forms the triangle $CMN$.

Piecing together slopes

I’m digging into the slopes of lines $OQ$ and $MN$ to see if their product is constant, considering geometric properties involving perpendicular bisectors and ellipse focal points.

Ensuring perpendicularity

Testing the slopes of lines $OQ$ and $MN$ to confirm their perpendicularity, aiming to determine if their product remains a constant value.

Solving intersections

I’ve identified intersections, slopes, and the midpoint of $MN$, and I’m working through equations to pinpoint how slopes $OQ$ and $MN$ intersect the ellipse.

Crunching numbers

I’m digging into the intersection of slopes $OQ$ and $MN$, finding their product as $-1$. This suggests $OQ$ is perpendicular to $MN$, and the focus is on the ellipse.

Analyzing intersections

I'm thinking through how the properties of slopes $OQ$ and $MN$ confirm their intersection, leading to $OQ$ being perpendicular to $MN$ and ensuring the focus is on the ellipse.

Thought about 曲线轨迹与几何性质，持续 5m 28s