Question

A gardener wants three rosebushes in her garden to be watered by a rotating water sprinkler. The gardener draws a diagram of the garden using a grid in which each unit represents 1 ft. The rosebushes are at (1, 3), (5, 11), and (11, 4). She wants to position the sprinkler at a point equidistant from each rosebush. Where should the gardener place the sprinkler? What equation describes the boundary of the circular region that the sprinkler will cover?

Answer 1

x^2 + y^2 + Ax + By + C = 0. Equation of a circle. You have 3 points, so you can get 3 equations in 3 unknowns (A, B, and C). Solvable linear system. Just substitute.

Answer 2

After you solve the linear system and have your equation: If you complete the square, the equation: x^2 + Ax + A^2/4 + y^2 + By + B^2/4 = -C + A^2/4 + B^2/4 = (x + A/2)^2 + (y + B/2)^2, so center at (-A/2, -B/2) with radius of sqrt(-C + A^2/4 + B^2/4)

Answer 3

There's a bit of work ahead of you, but the outline of what you have to do is all there. I can give you a little more detail. You take x^2 + y^2 + Ax + By + C = 0 and substitute the first (x, y) pair into it. Write it out. Do the same for the other 2 pairs. Write those 2 out and now you have 3 equations in 3 unknowns. You have to solve for A, B, C. Now you have ONE equation which all three points will satisfy. My second post gives you the center of the circle in terms of A and B, so that's where to put the sprinkler. The "reach" is the radius. It will end at the center of each rose bush.

Answer 4

thankyou for all ur help!!!!!!!!!!!! :D

Answer 5

You're welcome!