Find the equation of the locus of the point which moves so that its distance from the point (2,0) is 2/3 its distance from the line y=5.
see related questions below
Find the equation of the locus of a point that moves so that its distance from the point (2,8) is always equal to 5.
To find the equation of the locus of the point, we can follow these steps:
Step 1: Define the variables:
Let (x, y) be the coordinates of the point on the locus.
Step 2: Find the distance between the point (x, y) and the point (2, 0):
The distance between two points (x₁, y₁) and (x₂, y₂) is given by the distance formula:
d₁ = sqrt((x₂ - x₁)² + (y₂ - y₁)²)
In this case, we have:
d₁ = sqrt((x - 2)² + (y - 0)²)
Step 3: Find the distance between the point (x, y) and the line y = 5:
The distance between a point (x₁, y₁) and a line Ax + By + C = 0 is given by the formula:
d₂ = |Ax₁ + By₁ + C| / sqrt(A² + B²)
In this case, the equation of the line y = 5 can be written as 0x + 1y - 5 = 0, so we have:
d₂ = |0x + 1y - 5| / sqrt(0² + 1²)
Step 4: Set up the equation based on the given condition:
We know that the distance from the point (x, y) to (2, 0) is 2/3 of the distance to the line y = 5. Mathematically, we can express this as:
d₁ = (2/3) * d₂
Substituting the formulas from Steps 2 and 3:
sqrt((x - 2)² + y²) = (2/3) * (|0x + 1y - 5| / sqrt(1²))
sqrt((x - 2)² + y²) = (2/3) * |y - 5|
Since we have a square root on both sides of the equation, we can square both sides to eliminate it:
(x - 2)² + y² = (2/3)² * (y - 5)²
(x - 2)² + y² = (4/9) * (y - 5)²
Expanding and rearranging terms:
(x - 2)² - (4/9) * (y - 5)² + y² = 0
(x² - 4x + 4) - (4/9) * (y² - 10y + 25) + y² = 0
x² - 4x + 4 - (4/9)y² + (40/9)y - (100/9) + y² = 0
x² - 4x + y² - (4/9)y² + (40/9)y - (100/9) = 0
Simplifying and combining like terms, we get the equation of the locus:
x² - 4x + (5/9)y² - (40/9)y + (100/9) = 0
Therefore, the equation of the locus is x² - 4x + (5/9)y² - (40/9)y + (100/9) = 0.
To find the equation of the locus of a point that satisfies a certain condition, we need to determine the relationship between the coordinates of the point. In this case, we are looking for the relationship between the distance of the point from the point (2, 0) and its distance from the line y = 5.
Let's denote the coordinates of the moving point as (x, y). The distance between two points (x1, y1) and (x2, y2) can be found using the distance formula:
d = √((x2 - x1)^2 + (y2 - y1)^2)
In our case, the distance from the point (x, y) to the point (2, 0) is given as 2/3 times its distance from the line y = 5. We can express this mathematically as:
√((x - 2)^2 + y^2) = (2/3) * |y - 5|
To remove the absolute value, we can rewrite the equation as two separate equations, considering both the positive and negative values of (y - 5):
1. √((x - 2)^2 + y^2) = (2/3) * (y - 5), when y > 5
2. √((x - 2)^2 + y^2) = -(2/3) * (y - 5), when y < 5
Squaring both sides of each equation will help eliminate the square root:
1. (x - 2)^2 + y^2 = ((2/3) * (y - 5))^2, when y > 5
2. (x - 2)^2 + y^2 = ((-2/3) * (y - 5))^2, when y < 5
Expanding the equations:
1. (x^2 - 4x + 4) + y^2 = (4/9)(y^2 - 10y + 25), when y > 5
2. (x^2 - 4x + 4) + y^2 = (4/9)(y^2 - 10y + 25), when y < 5
Combining like terms:
1. (x^2 - 4x + 4) - (4/9)y^2 + (40/9)y - (100/9) = 0, when y > 5
2. (x^2 - 4x + 4) - (4/9)y^2 + (40/9)y - (100/9) = 0, when y < 5
Simplifying further:
1. (9x^2 - 36x + 36) - 4y^2 + (40/9)y - (100/9) = 0, when y > 5
2. (9x^2 - 36x + 36) - 4y^2 + (40/9)y - (100/9) = 0, when y < 5
Combining both cases into a single equation with the ± symbol for y:
(9x^2 - 36x + 36) - 4y^2 + (40/9)y - (100/9) = 0
Simplifying further, we can multiply the equation by 9 to eliminate fractions:
9(9x^2 - 36x + 36) - 36y^2 + (40)y - (100) = 0
Finally, rearranging the terms, we get the equation of the locus:
9x^2 - 36x - 36y^2 + 40y - 100 = 0