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.

1s= x^2+y^=10^2
3s= x^2+y^=30^2
10s= x^2+y^=100^2

is this correct.

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

x^2 + y^2 = r^2
-9^2+12^2=r^2
-81+144=r^2
7.94=r

is this right?

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

this one i'm stuck..

could you help me?

4a)

Let the point P(x,y) be a general point on the locus as described

the AP = BP
√[(x+2)^2 + (y-3)^2] = 2√[x-4)^2+(y+6)^2]

square both sides and expand, looks like a circle equation to me

for b) translate the given condition into a mathematical equation again using a general point P
isn't the slope(AP) equal to -slopeBP) ??

for 5a) you forgot the exponent of 2 on the y term, should have been y^2, probably just a typo

for 5c)
how long is the radius when you reach (-9,12) ?
find that length, divide it by 10 cm/s

a) To find the equation of the locus of the dog, we need to determine the set of points that satisfy the condition of being twice as far from A as they are from B.

Let's assume the dog is at a point (x, y). The distance between the dog and point A is given by the distance formula:
√((x - (-2))^2 + (y - 3)^2)

The distance between the dog and point B is given by:
√((x - 4)^2 + (y - (-6))^2)

Now, we need to set up the equation for the locus of the dog using the given condition that the distance from A is twice the distance from B:
√((x - (-2))^2 + (y - 3)^2) = 2 * √((x - 4)^2 + (y - (-6))^2)

To simplify this equation, we can square both sides:
(x - (-2))^2 + (y - 3)^2 = 4 * ((x - 4)^2 + (y - (-6))^2)

Expanding and rearranging terms:
(x + 2)^2 + (y - 3)^2 = 4 * ((x - 4)^2 + (y + 6)^2)

This is the equation of the locus of the dog. You can plot this equation on a graph to visualize the two trees, the path of the dog, and the relationship defining the locus.

b) To find the equation of the locus of the cat, we need to determine the set of points where the line segments between the cat and the two trees are always perpendicular.

Let's assume the cat is at a point (x, y). The vector from the cat to point A is given by:
<x - (-2), y - 3> = <x + 2, y - 3>

The vector from the cat to point B is given by:
<x - 4, y - (-6)> = <x - 4, y + 6>

For these two vectors to be perpendicular, their dot product must be zero:
(x + 2)(x - 4) + (y - 3)(y + 6) = 0

Expanding and simplifying:
x^2 - 2x - 8 + y^2 + 3y - 18 = 0
x^2 + y^2 - 2x + 3y - 26 = 0

This is the equation of the locus of the cat. You can plot this equation on a graph to visualize the path of the cat.

c) To determine how many seconds it takes for the ripple to reach point (-9, 12), we need to find the time it takes for the radius of the circular ripple to reach the distance between the origin (0,0) and (-9, 12).

The distance between the origin and (-9, 12) is given by the distance formula:
√((-9 - 0)^2 + (12 - 0)^2) = √(81 + 144) = √(225) = 15

Since the rate of increase of the radius is 10 cm/s, it will take 15 cm / 10 cm/s = 1.5 seconds for the ripple to reach point (-9, 12).