Conic Sections

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

If OO is the origin and OPOP and OQOQ are the two tangent lines from the origin to the circle x2+y210x+4y+6=0,x^2+y^2-10x+4y+6=0, what are the coordinates of the circumcenter of triangle OPQ?OPQ ?

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

For the two sets A={(x,y)x2+y2=9}, B={(0,1),(k,2)}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={(a1+b1,a2+b2)(a1,a2)A,(b1,b2)B}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 22 circles. If these 22 circles are externally tangential to each other, what is the value of k2k^2?


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