Several conics problems and solutions to questions on a quiz we did.

1. The foci of a hyperbola are F1(6, 0) and F2(-6, 0), and the difference of the
focal radii is 4 units. Use the definition of a hyperbola to derive the equation of
this hyperbola.

2. For each ellipse whose equation is given below
(i) Write the equation in standard form
(ii) Determine the lengths of the major and minor axes, the coordinates
of the verticies, and the coordinates of the foci.
(iii) Sketch a graph of the ellipse.

3. Determine the equaiton of a parabola defined by the given conditions.
The vertex is V(-1, 3) and the equation of the directrix is x - 2 = 0

4. A rock is kicked off a vertical cliff and falls in a
parabolic path to the water below. The cliff is 40 m
high and the rock hits the water 10 m from the base of
the cliff. What is the horizontal distance of the rock
from the cliff face when the rock is at a height of 30 m
above the water?

5. A hyperbola has centre (0, 0) and one vertex A .
(a) Find the equation of the hyperbola if it passes through J(9, 5)

6. A hyperbola has centre (0, 0) and one vertex A .
(a) Find the equation of the hyperbola if it passes through J(9, 5)
(c) Find the value of b if L(3, b)
(b) Find the value of a if K(a, 2)
is on the hyperbola.
is on the hyperbola.

7. Consider the parabola y2 - 20x + 2y + 1 = 0. Write the equation in standard
form, find the coordinates of the vertex and focus and the equation of the
directrix.

8. Write the equation of the circle x2 + y2 + 2x - 10y + 25 = 0 in standard form
and find the centre and the radius.

9. The parabola y2 - x + 4y + k = 0 passes through the point (12, 1). Find the
coordinates of the vertex and sketch the graph.

10. A point P(x, y) moves such that it is always equidistant from the point A(2, 3) and
the line y = -1. Determine the equation of this locus in standard form.

11. An ellipse has centre (-2, 4) and one vertex A(8, 4).
(a) Find the equation of the ellipse if it passes through R(4, 8).
(b) Find the value of c if S(c, 7) is on the ellipse.
(c) Find the value of d if T(3, d) is on the ellipse.