A point moves such that its distance from point A(1,0) is twice its distance from B(-2,0).Find the equation of the locus of the point.
let the point be P(x,y)
then PA = 2PB
√( (x-1)^2 + y^2) = 2√( (x+2)^2 + y^2)
square both sides and expand
x^2 - 2x + 1 + y^2 = 4(x^2 + 4x + 4 + y^2)
x^2 - 2x + 1 + y^2 = 4x^2 + 16x + 16 + 4y^2
3x^2 + 18x + 3y^2 = -15
x^2 + 6x + y^2 = -5
complete the square:
x^2 + 6x + 9 + y^2 = -5+9
(x+3)^2 + y^2 = 4
circle with centre (-3,0) and radius 2
confirmed here:
http://www.wolframalpha.com/input/?i=plot+√%28+%28x-1%29%5E2+%2B+y%5E2%29+%3D+2√%28+%28x%2B2%29%5E2+%2B+y%5E2%29
To find the equation of the locus of the point, we need to determine the relationship between the coordinates of the moving point and express it as an equation.
Let's assume the coordinates of the moving point are (x, y). We need to find the equation that satisfies the condition that the distance between the point (x, y) and point A(1, 0) is twice the distance between the point (x, y) and point B(-2, 0).
Step 1: Find the distance between the points (x, y) and A(1, 0).
The distance between two points (x1, y1) and (x2, y2) is given by the distance formula:
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In our case, the distance between (x, y) and A(1, 0) is:
Distance1 = sqrt((x - 1)^2 + (y - 0)^2) = sqrt((x - 1)^2 + y^2)
Step 2: Find the distance between the points (x, y) and B(-2, 0).
Similarly, the distance between (x, y) and B(-2, 0) can be calculated as:
Distance2 = sqrt((x - (-2))^2 + (y - 0)^2) = sqrt((x + 2)^2 + y^2)
Step 3: Set up the equation based on the given condition.
According to the problem, the distance from A to (x, y) is twice the distance from B to (x, y). Mathematically, we can express this as:
Distance1 = 2 * Distance2
Replacing the distance formulas from steps 1 and 2:
sqrt((x - 1)^2 + y^2) = 2 * sqrt((x + 2)^2 + y^2)
To simplify the equation, we can square both sides:
(x - 1)^2 + y^2 = 4 * ((x + 2)^2 + y^2)
Expand and simplify:
x^2 - 2x + 1 + y^2 = 4 * (x^2 + 4x + 4 + y^2)
x^2 - 2x + 1 + y^2 = 4x^2 + 16x + 16 + 4y^2
Rearranging the equation:
3x^2 + 14x + 3y^2 - 15 = 0
Therefore, the equation of the locus of the point is 3x^2 + 14x + 3y^2 - 15 = 0.