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.
To find the equation of the curve traced by point P(x, y), we need to express the relationship described in the given problem as an equation. Let's break down the problem statement.
We're told that the distance from point P to point A is three times the distance from point P to point B. We can represent these distances using the distance formula.
The distance between two points (x₁, y₁) and (x₂, y₂) is given by:
Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
Let's use this formula to express the given information in terms of the distances between P, A, and B.
Distance from P to A:
√((x - (-1))² + (y - 1)²) = d₁
Distance from P to B:
√((x - 2)² + (y - (-1))²) = d₂
According to the problem, d₁ (distance from P to A) is three times d₂ (distance from P to B). We can write this relationship as an equation:
d₁ = 3d₂
Substituting the expressions for d₁ and d₂, we get:
√((x - (-1))² + (y - 1)²) = 3 * √((x - 2)² + (y - (-1))²)
Squaring both sides of the equation eliminates the square root:
((x - (-1))² + (y - 1)²) = 9 * ((x - 2)² + (y - (-1))²)
Expanding the equations and simplifying, we get:
(x + 1)² + (y - 1)² = 9 * [(x - 2)² + (y + 1)²]
Now, let's simplify this equation to get the final form of the equation of the curve traced by point P.
(x + 1)² + (y - 1)² = 9 * (x - 2)² + 9 * (y + 1)²
(x² + 2x + 1) + (y² - 2y + 1) = 9x² - 36x + 36 + 9y² + 18y + 9
Simplifying further:
x² + y² + 2x - 2y + 2 = 9x² - 36x + 9y² + 18y + 45
Rearranging terms and combining like terms:
8x² - 38x + 8y² + 20y + 43 = 0
Thus, the equation of the curve traced by point P is:
8x² + 8y² - 38x + 20y + 43 = 0
To identify the specific type of curve represented by this equation, we can rewrite it in standard form.
8x² + 8y² - 38x + 20y + 43 = 0
Divide through by the constant term on the right side to simplify the equation:
x² + (38/8)x + y² + (20/8)y + (43/8) = 0
x² - (19/4)x + y² + (5/2)y + (43/8) = 0
Completing the square for x and y terms separately:
(x² - (19/4)x + (19/8)²) + (y² + (5/2)y + (5/4)²) = (19/8)² + (5/4)² - (43/8)
(x - (19/8))² + (y + (5/4))² = (361/64) + (25/16) - (344/64)
(x - (19/8))² + (y + (5/4))² = (361 + 100 - 344) / 64
(x - (19/8))² + (y + (5/4))² = 117 / 64
Comparing this equation with the standard form of a circle equation:
(x - h)² + (y - k)² = r²
We see that the equation represents a circle centered at (h, k) = (19/8, -5/4) with a radius of √(117/64).
Therefore, the equation of the curve traced by point P is a circle with center (19/8, -5/4) and radius √(117/64).
To find the equation of the curve traced by the point P(x, y), we will use the distance formula. The distance between two points A and B in the coordinate plane is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Let's consider the distance from P to A and B: PA and PB, respectively.
According to the problem, the distance from P to A is three times the distance from P to B. Mathematically, this can be expressed as:
PA = 3 * PB
Using the distance formula, we can calculate PA and PB:
PA = sqrt((x - (-1))^2 + (y - 1)^2)
= sqrt((x + 1)^2 + (y - 1)^2)
PB = sqrt((x - 2)^2 + (y - (-1))^2)
= sqrt((x - 2)^2 + (y + 1)^2)
We can rewrite the given equation as:
(sqrt((x + 1)^2 + (y - 1)^2)) = 3 * (sqrt((x - 2)^2 + (y + 1)^2))
Now, we can square both sides of the equation to eliminate the square roots:
((x + 1)^2 + (y - 1)^2) = 9 * ((x - 2)^2 + (y + 1)^2)
Expanding both sides of the equation:
(x^2 + 2x + 1 + y^2 - 2y + 1) = 9 * (x^2 - 4x + 4 + y^2 + 2y + 1)
Simplifying the equation:
x^2 + 2x + y^2 - 2y + 2 = 9x^2 - 36x + 36 + 9y^2 + 18y + 9
Rearranging the terms:
8x^2 - 38x + 8y^2 + 16y - 43 = 0
Thus, the equation of the curve traced by the point P(x, y) is:
8x^2 - 38x + 8y^2 + 16y - 43 = 0
To identify the type of curve, we can examine the coefficients of x^2 and y^2. Since both are positive and the coefficients of x^2 and y^2 are different, the curve represents an ellipse.