1.
Item 3. State the coordinates of the center for the following conic: \(16(x+1)^2+(y-2)^2=64\).
Solution: The center of a conic in translated form is found from the expressions \((x-h)^2\) and \((y-k)^2\). Here \(x+1=x-(-1)\), so \(h=-1\). Also \(y-2\) gives \(k=2\). Therefore the center is \((-1,2)\). Answer: B
2.
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 further away from the vertex.
Solution: When the plane remains parallel to its original position and moves further away from the vertex of the cone, the circular cross-section becomes larger. Therefore, the radius of the circle increases and the circle is larger. Answer: A
3.
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 \(x^2+y^2=r^2\). Since \(r=\sqrt{5}\), \(r^2=(\sqrt{5})^2=5\). Therefore the equation is \(x^2+y^2=5\). Answer: B
4.
Item 4. What is the equation of the following parabola?
Solution: The graph shows a parabola with a horizontal axis of symmetry and vertex \((2,3)\). A horizontally opening parabola has standard form \((y-k)^2=4a(x-h)\). Substituting \(h=2\) and \(k=3\) gives \((y-3)^2=4a(x-2)\). Answer: B
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