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 + ...

Solving parabola problem

I need to solve this in Chinese, focusing on a parabola with fixed focal chord AB, where A(-1,0) and B(1,0), and the directrix is tangent to the circle x^2 + y^2 = 9. The focus P moves, and I'm asked to find the equation of curve E formed by the set of all such foci. The directrix is tangent to the circle, and I need to find the relationship between the coordinates of the focus. The focus's position involves using the definition of a parabola: the distance to the focus equals the distance to the directrix.

Considering distances and focus

We have two fixed points A(-1,0) and B(1,0) on the parabola. For a point P (the focus) and a directrix L, the distance from A to P equals the distance from A to L, and similarly for B. I’m thinking of using the equidistant property of parabolas and the relationship between focus and directrix to determine how A and B connect to the focus. The directrix L is tangent to the circle x^2 + y^2 = 9, and we know that this distance is 3. This can help solve the problem.

Deriving focus and directrix relations

I’m working through relationships involving the focus and directrix of the parabola. For A(-1,0) and B(1,0), the distance from A and B to the directrix involves calculating absolute values based on the equation of the line. When we subtract these distances, I find a relation between the coefficients a and the focus p. After some more steps, it seems the focus lies on the y-axis due to the symmetry of A and B. Contradictions suggest corrections, and I suspect the focus is at (0, q), but the actual solution depends on testing specific cases.

Locus of the focus

I'm finding the locus of the focus for the parabola with directrices at y = 3 or y = -3, and passing through A(-1, 0) and B(1, 0). After some calculations, I derive that the locus of foci satisfies the ellipse equation $\frac{x^2}{9} + \frac{y^2}{8} = 1$. The focus points lie on this ellipse, with $p^2 + q^2$ between 8 and 9. So, the answer is an ellipse, and I confirm the equation matches the necessary conditions.

Finding circumcenter Q

I'm exploring how to calculate the circumcenter Q for triangle CMN, where points M and N are on the intersection of the ellipse and a line. I want to find the relationship between the coordinates of Q, the midpoint of M and N, and the fixed point C. To do this, I use the fact that Q is equidistant from C and any M or N. I’m trying to eliminate unnecessary variables, and I'm also considering using certain geometrical properties like the radical axis to simplify the process.

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