1.
Item 1. A plane intersects a double-napped cone to form a circle. Assume the plane moves parallel to its original position. Describe what happens to the circle that is formed when the plane moves closer to the vertex.
Solution: When a plane cuts a double-napped cone parallel to its original position and moves closer to the vertex, the circular cross-section becomes smaller. Therefore the radius of the circle decreases, so the circle is smaller. Answer: A
2.
Item 2. What is the equation of a circle with a center at \((0,0)\) and a radius of \(\sqrt{5}\)?
Solution: The standard equation of a circle centered at \((0,0)\) is \(x^2+y^2=r^2\). Here \(r=\sqrt{5}\), so \(r^2=(\sqrt{5})^2=5\). Therefore the equation is \(x^2+y^2=5\). Answer: D
3.
Item 3. Which equation represents the following graph?
Solution: From the graph, the ellipse has center \((-3,1)\). In standard form, an ellipse centered at \((h,k)\) contains \((x-h)^2\) and \((y-k)^2\). Since \(h=-3\) and \(k=1\), the equation must contain \((x+3)^2\) and \((y-1)^2\). The graph is wider horizontally, so the larger denominator \(9\) is under the \(x\)-term and \(4\) is under the \(y\)-term. Therefore the equation is \(\frac{(x+3)^2}{9}+\frac{(y-1)^2}{4}=1\). Answer: A
4.
Item 4. What is the equation for the following conic: A parabola with vertex at \((-3,2)\) and a horizontal axis of symmetry.
Solution: A parabola with a horizontal axis of symmetry has standard form \((y-k)^2=4a(x-h)\), where \((h,k)\) is the vertex. The vertex is \((-3,2)\), so \(h=-3\) and \(k=2\). Substitute into the formula: \((y-2)^2=4a(x-(-3))=4a(x+3)\). Answer: C
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