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Conic Sections

Parabolas, ellipses, and hyperbolas, oh my! Learn about this eccentric bunch of shapes.

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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