Loci and Circles

1. Describe each locus algebraically. Then graph the locus and describe the locus geometrically. For part (d), prove that the locus is the perpendicular bisector of the line joining the two points.

a) locus of points 5 units above or below the x-axis

b) locus of points equidistant from (-3,0) and (5,0)

c) locus of points 10 units from (-3,4)

d) locus of points equidistant from (0,0) and (-6,-6)

2. Consider the circle (x+5)squared+(y+4)squared=20 and the line y=2x+1.

a) State the centre and the radius of the circle.

b) Draw a graph of the circle and the line.

c) Is the line a tangent or a secant to the circle?

d) Approximate the coordinates of the points of intersection using the graph.

e) Find the points of intersection of the circle and the line algebraically (to two decimal points).

3. Consider the circle with equation xsquared-4x+ysquared-10y-25=0.

a) Express the equation of the circle in standard form.

b) State the radius and the centre of the circle.

c) Draw a graph of the circle.

d) Find the x-intercepts and y-intercepts of the circle.

e) Describe, algebraically, the translation that will move the circle so that its centre is at the origin.

4. In a backyard, there are two trees located at grid points A(-2,3) and B(4,-6).

a) The family dog is walking through the backyard so that it is at all times twice as far from A as it is from B. Find the equation of the locus of the dog. Draw a graph that shows the two trees, the path of the dog, and the relationship defining the locus. Then write a geometric description of the path of the dog relative to the two trees.

b) The family cat is also walking in the backyard. The line segments between the cat and the two trees are always perpendicular. Find the equation of the locus of the cat. Draw a graph that shows the path of the cat. Then write a geometric description of the path of the cat relative to the two trees.

5. A pebble is thrown into a pond at a point that can be considered the origin, (0,0). Circular ripples move away from the origin such that the radius of the circle increases at a rate of 10cm/s.

a) State the equations of the ripples after 1s, after 3s, and after 10s.

b) Descibe the equation of the circle that contains point (-9,12)?

c) How many seconds does the ripple take to reach point (-9,12)?