已知过点 $A(-1, 0)$ 、 $B(1, 0)$ 两点的动抛物线的准线始终与圆 $x^2 + ...

Translating the math problem

Alright, so this problem involves a family of parabolas passing through points A(-1,0) and B(1,0). Their directrix is tangent to the circle $x^2 + y^2 = 9$. We're asked to find the standard equation of the conic traced by the parabolas' focus P.

Plus, part (2) introduces points like C(-3,0) and D(2,0), and asks about relationships involving certain lines and slopes — a triangle with its circumcenter at Q, and the origin O.

I need to break this into steps and solve!

Solving the slopes' product

So, first things first, I need to figure out if the product of the slopes of lines OQ and MN is constant. But before that, the question starts with part (1), which involves solving for a parabola defined by points A(-1,0) and B(1,0) and focus P.

The parabola is a variable one with a focus and directrix, and the directrix is tangent to the circle $x^2 + y^2 = 9$. I’m realizing the challenge is relating the parabola’s properties to the equation of the focus, P, and determining how these curves move!

Okay, let me keep working through this.

Determining directrix and focus

Okay, let's break this down. We're looking for the directrix of a parabola passing through A(-1,0) and B(1,0), with its focus (0, q) on the y-axis. I deduced that the directrix is horizontal, in the form $y = k$, where $k = 3$ or $k = -3$.

For symmetry, the focus and directrix lie on opposite sides of the vertex. Since the parabola passes through A and B, the vertex is likely between the focus and the directrix. So, I'll choose $k = -3$. This makes sense with the given tangency condition to the circle $x^2 + y^2 = 9$.

Re-examining the locus of the focus

So, I realized something important: the directrix for the moving parabolas is tangent to the circle $x^2 + y^2 = 9$. This introduces some complexity because the directrix doesn't have to be horizontal—it can have other orientations, affecting the location of the focus.

When I worked out the focus was (0, $\pm 2\sqrt{2}$), I thought that was it, but the focus may trace a curve, not just isolated points. So the locus of the focus could be part of a conic, not just fixed points. I'm not done yet!

Reviewing possible parabolas

I’ve identified two possibilities for the focus based on whether $k = 3$ or $k = -3$. The condition from point A shows that for both cases, $q = \pm 2\sqrt{2}$, and I found both values are valid. However, the sign of $q$ impacts which side of the vertex the focus lies on. The vertex must be between the directrix and the focus—this determines how the parabola opens. So, I now have two possible parabolas, each with a different focus but satisfying the given conditions. Both orientations work!

Reassessing the locus of the focus

When I revisit everything, the focus seems to be constrained between two fixed points depending on different value allocations. This gives only isolated points instead of a conic curve, which contradicts what was originally posed as a moving parabola. But it’s making more sense now that the focus traces an ellipse. Through careful rework, the equation of the ellipse $\frac{x^2}{9} + \frac{y^2}{8} = 1$ emerges, providing the correct locus. This ellipse fits correctly with the conditions set in the problem.

Finding the circumcenter

To find the circumcenter Q, I need to approach the problem in manageable steps. First, I determine the midpoint of M and N, points of intersection, using their x and y coordinates. Then, I focus on the perpendicular bisector of segment MN, which has a slope of -1/k. After that, the circumcenter Q lies at the intersection of this bisector and the bisector of CM or CN. The key point is that Q is equidistant from both C and the points M, N, which simplifies several calculations. I can proceed with this carefully.

Finding circumcenter Q

To find the coordinates of the circumcenter Q, I need to consider the relationship between the triangle vertices C, M, and N. Q is equidistant from C, M, and N. The approach involves solving the perpendicular bisectors of segments CM and CN, using coordinates for M and N derived from the line L intersecting the ellipse. I explore this through geometric relations, using midpoint calculations, slopes, and substituting into the equation of a circle. Ultimately, I'm concluding that the circumcenter is computable using formulas tied to these geometries.

Determining the product of slopes

I'm trying to determine if the product between the slope of OQ and MN is constant. After some calculations, I've found that for various values of k, like k = 1 and k = 2, the product of the slopes is approximately -5.02 and -5.4, respectively. When examing it further symbolically, I find that the product seems to stabilize around -5, regardless of the value of k. Therefore, it appears that the constant value for the product of slopes is approximately -5.

推理关于抛物线与圆的关系 for 1m 44s