For each point in a set of points, its distance from (-2,4) is three times its distance from the line x=6
1. find the equation
2. tell which conic section the graph will be.
looks like a parabola.
distance from (x,y) to x=6 is x-6
distance from (-2,4) given by (x+2)^2 + (y-4)^2
vertex at (4,4) since it is 2 from x=6 and 6 from (-2,4)
(3(x-6))^2 = (x+2)^2 + (y-4)^2
or
9(x-6)^2 - (x+2)^2 = (y-4)^2
Not sure how to reduce that to
(y-4) = a(x-4)^2
1. Let's denote a general point in the set of points as (x, y). According to the given conditions, we have the following two equations:
Distance from (-2, 4) = 3 * Distance from x = 6
√((x - (-2))^2 + (y - 4)^2) = 3 * |x - 6|
Simplifying the equation:
√((x + 2)^2 + (y - 4)^2) = 3 * |x - 6|
Squaring both sides to eliminate the square root:
(x + 2)^2 + (y - 4)^2 = 9(x - 6)^2
Expanding and rearranging, we get:
x^2 + 4x + 4 + y^2 - 8y + 16 = 9x^2 - 108x + 324
Simplifying further:
8x^2 - 112x + 140 + y^2 - 8y = 0
2. By looking at the equation, we can observe that the degrees of x and y terms are both 2, and the coefficients of those terms have opposite signs. This indicates that the graph will represent an elliptical conic section.
To find the equation of the graph and determine which conic section it represents, we can follow these steps:
1. Find the equation:
Let's consider a general point in the set of points as (x, y).
According to the given information, the distance between the point (x, y) and (-2, 4) is three times the distance between the point (x, y) and the line x = 6.
Using the distance formula, the distance between two points (x1, y1) and (x2, y2) is given by:
d = √((x2 - x1)^2 + (y2 - y1)^2)
In this case, the distances are given by:
√((x - (-2))^2 + (y - 4)^2) = 3 * √((x - 6)^2 + (y - 0)^2)
Squaring both sides of the equation, we get:
(x + 2)^2 + (y - 4)^2 = 9 * ((x - 6)^2 + y^2)
Expanding and simplifying the equation:
x^2 + 4x + 4 + y^2 - 8y + 16 = 9x^2 - 108x + 324 + 9y^2
This equation can be further simplified to:
8x^2 + 113x + 9y^2 + 8y - 220 = 0
Therefore, the equation of the graph is 8x^2 + 113x + 9y^2 + 8y - 220 = 0.
2. Identify the conic section:
By examining the equation, we can determine the conic section based on its form. In this case, the equation is in the form of:
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
By comparing the coefficients, we can see that the equation represents an ellipse. This is because both x^2 and y^2 appear with the same sign (positive), indicating that it is not a hyperbola. Additionally, the coefficients A and C are both positive, which further confirms that it is an ellipse.
To find the equation and determine the conic section of the graph, we need to use the distance formula and apply the given conditions. Let's break down the steps:
1. Let's assume the coordinates of the point in the set are (x, y). According to the given condition, the distance between this point and (-2, 4) is three times its distance from the line x = 6.
Using the distance formula, the distance between two points (x₁, y₁) and (x₂, y₂) is given by:
d = √[(x₂ - x₁)² + (y₂ - y₁)²]
Therefore, the equation based on the given condition is:
√[(x - (-2))² + (y - 4)²] = 3 * √[(x - 6)² + (y - 0)²]
Simplifying this equation, we get:
√[(x + 2)² + (y - 4)²] = 3 * √[(x - 6)² + y²]
Squaring both sides of the equation, we obtain:
(x + 2)² + (y - 4)² = 9 * [(x - 6)² + y²]
Expanding and rearranging the terms gives us the equation:
25x² - 92x - 45y² + 52y + 300 = 0
Therefore, the equation of the graph is: 25x² - 92x - 45y² + 52y + 300 = 0.
2. To determine the conic section, we can analyze the equation. By examining the terms, we can identify the values of the coefficients in front of x², y², and the xy terms.
In this case, we have:
- Coefficient of x² (25) ≠ Coefficient of y² (-45)
- Coefficient of xy (0)
Since the coefficients of x² and y² have different signs, and the coefficient of xy is zero, we can conclude that the graph represents an ellipse.