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 is 1:2
To find the location of Alonso's home, we need to calculate the distances from both the animal shelter and the grooming service.
Let Alonso's home be located at (x, y) on the coordinate plane.
The distance from the shelter to Alonso's home is given as 1 part, and the distance from Alonso's home to the grooming service is given as 2 parts.
Using the distance formula between two points, we have:
Distance from shelter to Alonso's home:
√[(x - (-1))^2 + (y - 9)^2] = √[(x + 1)^2 + (y - 9)^2]
Distance from Alonso's home to grooming service:
√[(x - 11)^2 + (y - 0)^2] = √[(x - 11)^2 + y^2]
According to the ratio, the distance from the shelter to Alonso's home is 1 part, and the distance from Alonso's home to the grooming service is 2 parts:
√[(x + 1)^2 + (y - 9)^2] : √[(x - 11)^2 + y^2] = 1 : 2
Squaring both sides of the equation to eliminate the square roots:
[(x + 1)^2 + (y - 9)^2] : [(x - 11)^2 + y^2] = 1 : 4
Multiply both sides by [(x - 11)^2 + y^2] to get rid of the denominator:
(x + 1)^2 + (y - 9)^2 = 4[(x - 11)^2 + y^2]
Expanding:
x^2 + 2x + 1 + y^2 - 18y + 81 = 4x^2 - 88x + 484 + 4y^2
Rearranging terms:
3x^2 - 90x + 3y^2 + 18y + 402 = 0
At this point, we have the equation of an ellipse. To solve for x and y, more information is needed, such as additional equations or constraints.