Find the coordinates of the center of the ellipse represented by 4x^2 + 9y^2 – 18y – 27 = 0.
For 2x^2 + xy + 2y^2 = 1, find , the angle of rotation about the origin, to the nearest degree.
Find the distance between points at (–1, 6) and (5, –2).
Identify the conic section represented by x^2 – y^2 + 12y + 18x = 42.
Find the coordinates of the points of intersection of the graphs of x^2 + y^2 = 5, xy = –2, and y = –3x + 1.
(0,1)
45
36
To find the coordinates of the center of the ellipse represented by 4x^2 + 9y^2 – 18y – 27 = 0, we need to rewrite the equation in standard form, which is (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where (h,k) is the center of the ellipse.
Step 1: Simplify the equation
Start by rearranging the terms: 4x^2 + 9y^2 – 18y – 27 = 0
Group the x and y terms together: 4x^2 + 9(y^2 – 2y) = 27
Complete the square for the y terms: 4x^2 + 9(y^2 – 2y + 1) = 27 + 9(1)
Simplify: 4x^2 + 9(y - 1)^2 = 36.
Step 2: Divide both sides by 36
Divide both sides of the equation by 36 to simplify: (4x^2)/36 + (9(y - 1)^2)/36 = 1
Simplify: (x^2)/9 + (y - 1)^2/4 = 1
Now the equation is in standard form.
Step 3: Identify the center
Comparing the equation to the standard form, we can see that the center of the ellipse is at the coordinates (h, k). From the equation, we can see that h = 0 and k = 1. Therefore, the center is at (0, 1).
To find the angle of rotation about the origin for the equation 2x^2 + xy + 2y^2 = 1, we need to rewrite the equation in terms of rotated coordinates.
Step 1: Rewrite the equation in matrix form
The given equation can be written in matrix form as X^TAX = 1, where X = [x, y] and A is the matrix of coefficients.
Step 2: Factorize the matrix A
Factorize the matrix A into a product of three matrices: A = P^TDP, where P is the matrix of eigenvectors and D is a diagonal matrix.
Step 3: Compute the angle of rotation
The angle of rotation can be calculated using the formula: θ = arctan((2B)/(A - C)), where A, B, and C are the entries of the diagonal matrix D.
Step 4: Round the angle to the nearest degree
Finally, round the angle of rotation obtained from Step 3 to the nearest degree.
To find the distance between points at (-1, 6) and (5, -2), we can use the distance formula.
Step 1: Identify the coordinates of the given points
The first point has coordinates (-1, 6) and the second point has coordinates (5, -2).
Step 2: Apply the distance formula
The distance formula is given by: d = √((x₂ - x₁)^2 + (y₂ - y₁)^2), where (x₁, y₁) and (x₂, y₂) are the coordinates of the two points.
Step 3: Substitute the coordinates into the distance formula
Substitute the coordinates (-1, 6) and (5, -2) into the distance formula:
d = √((5 - (-1))^2 + (-2 - 6)^2).
Step 4: Simplify and compute the distance
Compute the distance by simplifying the expression under the square root sign, and then taking its square root.
To identify the conic section represented by x^2 – y^2 + 12y + 18x = 42, we can analyze the equation and determine its standard form.
Step 1: Rearrange the equation
Rearrange the given equation: x^2 + 18x – y^2 + 12y = 42.
Step 2: Complete the square for both x and y terms
Complete the square for the x terms by adding (18/2)^2 = 81 to both sides of the equation. Complete the square for the y terms by adding (12/2)^2 = 36 to both sides.
Step 3: Simplify the equation
Simplify the equation after completing the square:
(x + 9)^2 – (y – 6)^2 = 117.
Step 4: Analyze the standard form
The standard form of a conic section equation with a positive sign between the x and y terms is: (x – h)^2/a^2 – (y – k)^2/b^2 = 1.
The standard form of a conic section equation with a negative sign between the x and y terms is: (x – h)^2/a^2 – (y – k)^2/b^2 = -1.
Comparing the equation (x + 9)^2 – (y – 6)^2 = 117 to the standard forms, we can see that it represents a hyperbola.
To find the coordinates of the points of intersection of the graphs of x^2 + y^2 = 5, xy = -2, and y = -3x + 1, we need to solve the system of equations.
Step 1: Set up the system of equations
The given equations are x^2 + y^2 = 5, xy = -2, and y = -3x + 1.
Step 2: Substitute y in terms of x in the first equation
Substitute y = -3x + 1 into the equation x^2 + y^2 = 5 to obtain a quadratic equation in terms of x.
Step 3: Solve the quadratic equation
Solve the quadratic equation obtained from step 2 to find the x-values of the points of intersection.
Step 4: Substitute the x-values into the equation y = -3x + 1
Substitute the x-values obtained from step 3 into the equation y = -3x + 1 to find the corresponding y-values.
Step 5: Write the coordinates of the points of intersection
Write the coordinates of the points of intersection as (x, y) pairs obtained from step 4.