A curve is traced by a point P(x,y) which moves such that its distance from the point A(-1,1) is three times the distance from the point B(2,-1). Find the equation of the curve and identity.
distancePB=3distancePA
sqrt((x-2)^2+(y+1)^2)=3sqrt((x+1)^2 +(y-1)^2)
square both sides, expand the polynomial, collect terms, and you ahve the equation.
To find the equation of the curve traced by point P(x, y), we need to express the given condition in mathematical terms.
Let's call the distance from point P to point A as dA and the distance from point P to point B as dB.
According to the given condition, the distance from point P to point A is three times the distance from point P to point B. We can express this as:
dA = 3 * dB
Now, let's calculate the distances:
dA = sqrt((x - (-1))^2 + (y - 1)^2)
dB = sqrt((x - 2)^2 + (y - (-1))^2)
Substituting the expressions for dA and dB into the equation we derived earlier:
sqrt((x - (-1))^2 + (y - 1)^2) = 3 * sqrt((x - 2)^2 + (y - (-1))^2)
To simplify this equation, we can square both sides to eliminate the square roots:
[(x - (-1))^2 + (y - 1)^2] = (3 * sqrt((x - 2)^2 + (y - (-1))^2))^2
(x - (-1))^2 + (y - 1)^2 = 9 * ((x - 2)^2 + (y - (-1))^2)
Expanding and combining like terms:
(x + 1)^2 + (y - 1)^2 = 9 * [(x - 2)^2 + (y + 1)^2]
(x^2 + 2x + 1) + (y^2 - 2y + 1) = 9 * (x^2 - 4x + 4 + y^2 + 2y + 1)
Simplifying further:
x^2 + 2x + y^2 - 2y + 2 = 9x^2 - 36x + 36 + 9y^2 + 18y + 9
0 = 8x^2 - 38x + 8y^2 + 16y + 43
This equation represents the curve traced by point P. To determine its identity, we can analyze its general form:
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
Comparing this general form with the equation we derived:
8x^2 - 38x + 8y^2 + 16y + 43 = 0
We can identify A = 8, B = 0, C = 8, D = -38, E = 16, and F = 43.
In conclusion, the equation of the curve traced by point P(x, y) is 8x^2 - 38x + 8y^2 + 16y + 43 = 0, and its identity is an ellipse.