Alonso plans to go the animal shelter to adopt a dog and then take the dog precious pup grooming services the shelter is located at (-1,9) on the coordinate plane ,while precious pup grooming services is located at (11,0) on the coordinate plane. find the locust ion it Alonso’s home if the ratio of the distance or shelter to his house to the distance of his home to precious pup grooming
To find the location of Alonso's home, we need to calculate the coordinates that lie between the shelter and precious pup grooming services.
Let's assume Alonso's home is located at (x, y).
The distance between the shelter (-1, 9) and Alonso's home (x, y) is given by the distance formula: √((x - (-1))^2 + (y - 9)^2).
And, the distance between Alonso's home (x, y) and precious pup grooming services (11, 0) is given by the distance formula: √((x - 11)^2 + (y - 0)^2).
The ratio of the distance from the shelter to his home to the distance from his home to precious pup grooming services is 1:2. So, the distance from the shelter to Alonso's home is half the distance from Alonso's home to precious pup grooming services.
Therefore, we can write the equation:
√((x - (-1))^2 + (y - 9)^2) = 0.5 * √((x - 11)^2 + (y - 0)^2)
Squaring both sides of the equation to eliminate the square root:
(x - (-1))^2 + (y - 9)^2 = 0.25 * ((x - 11)^2 + (y - 0)^2)
Expanding both sides:
(x + 1)^2 + (y - 9)^2 = 0.25 * (x^2 - 22x + 121 + y^2)
(x^2 + 2x + 1) + (y^2 - 18y + 81) = 0.25x^2 - 5.5x + 30.25 + 0.25y^2
Rearranging terms:
3x^2 + 36x + 3y^2 - 54y - 107 = 0
We don't know the exact location of Alonso's home yet, but this equation represents all the possible coordinates (x, y) for his home that satisfy the given conditions.