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# Conic Sections - Problem Solving

What can we say about light rays that are shot out from the focus of a parabolic mirror, and reflected?

In the above diagram, a right cone with vertex $$A=(0,16,32)$$ and circular base with center $$O=(0,16,0)$$ is standing on the $$xy$$-plane such that the origin is an endpoint of a diameter of the circular base. If $$\alpha$$ is a plane which passes through both $$A$$ and $$O,$$ what is the area of the intersection between the cone and the plane $$\alpha?$$

If $$O$$ is the origin and $$OP$$ and $$OQ$$ are the two tangent lines from the origin to the circle $x^2+y^2-10x+4y+6=0,$ what are the coordinates of the circumcenter of triangle $$OPQ ?$$

The above diagram shows a portion of an infinite right cone and a circular cross section centered at $$O$$ that is perpendicular to the axis of the cone. Suppose $$\angle ABC=60^\circ$$ and $$\overline{MON}$$ is a diameter of the circular cross section. If the shape cut by a plane containing $$\overline{MON}$$ and a point $$D$$ on the cone surface is a parabola, what is the value of $$x$$ in degrees?

For the two sets $A=\left\{ (x,y) \mid x^2+y^2=9 \right\},\ B=\left\{ (0,1), (k,2) \right\}$ on the xy-plane, a set $P=\left\{ (a_1+b_1, a_2+b_2) \mid (a_1, a_2) \in A, (b_1, b_2) \in B \right\}$ represents $$2$$ circles. If these $$2$$ circles are externally tangential to each other, what is the value of $$k^2$$?

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