Solve for the standard equation of the given characteristics of each conic then graph. Show complete solution then graph and label all important characteristics of the conic properly.
1. Find the standard form of the equation of the circle tangent to both axes at (−3, 0) and (0, 3).
2. Find the standard equation of the parabola with directrix ? + 1 = 0, axis at ? = 1, and the length of the latus rectum is 4.
3. Discuss the graph of the ellipse with foci at (2, 3) and (2, −1) and an endpoint of the minor axis at (4, 1).
4. Find the standard equation of the hyperbola with the same vertices as the vertices of the ellipse 9?2 + 4?2 = 36 and with the asymptotes ? = ±3/2?.
5. Find the standard form of the equation of the circle, if the center of the circle of radius 3 lies on the line ? + 2? = 4 and it is tangent to the y – axis.
6. Show that the product of the distances of any point of a hyperbola from its asymptotes is constant.
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