10.6 Parametric Forms Of Equations

Explanation
The given equation x^2 - 4y - 6x + 9 = 0 can be rearranged into the standard form of a parabola, which is y = a(x-h)^2 + k. By completing the square, we can rewrite the equation as (x-3)^2 = 4(y-1), which matches the standard form of a parabola. Therefore, the conic described by the equation is a parabola.

Explanation
The given equation can be rearranged to the standard form of a quadratic equation, which is Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. In this case, the equation is x^2 - 4y - 6x + 9 = 0. By simplifying the equation, we get 4y = (x - 3)^2. Therefore, the correct answer is 4y = (x - 3)^2.

Explanation
The equation x^2 - 3y^2 + 2x - 24y - 41 = 0 is a hyperbola because it is a second-degree equation with both x^2 and y^2 terms, and the coefficients of x^2 and y^2 have opposite signs. This indicates that the graph of the equation will be a hyperbola.

Explanation
The given equation is in the standard form of a hyperbola, which is (y+4)^2 / 2 - (x+1)^2 / 6 = 1. This is determined by comparing the given equation with the general equation of a hyperbola, (x-h)^2 / a^2 - (y-k)^2 / b^2 = 1, where (h,k) represents the center of the hyperbola and a and b are the distances from the center to the vertices. In this case, the center is (-1,-4), a^2 = 6, and b^2 = 2. Therefore, the correct answer is (y+4)^2 / 2 - (x+1)^2 / 6 = 1.

Explanation
The given equations are parametric equations in terms of the parameter t. To convert them into rectangular equations, we can substitute x = t into the equation for y. By doing so, we get y = 2(t)^2 - 4(t) + 1, which simplifies to y = 2t^2 - 4t + 1. Therefore, the correct rectangular equation is y = 2x^2 - 4x + 1.

Explanation
The given parametric equations x = cos 2t and y = sin 2t represent a parametric curve in the xy-plane. By substituting these equations into the equation x^2 + y^2 = 1, we can verify that it holds true for all values of t between 0 and 2pi. This equation represents a unit circle centered at the origin, which means that any point (x, y) on the curve satisfies the equation x^2 + y^2 = 1. Therefore, the correct answer is x^2 + y^2 = 1.

Explanation
The given parametric equations x = -cos t and y = sin t represent a circle centered at the origin with radius 1. In the answer choice x = cos 2t and y = sin 2t, the parametric equations represent the same circle, but with the parameter t multiplied by 2. Therefore, the rectangular equation for the given problem is x = cos 2t and y = sin 2t.

Explanation
The explanation for the given correct answer is that the statement is incorrect. In the general form of a conic, a parabola can be identified if and only if the x term is squared and the y term is not squared. Therefore, the correct answer is false, as it contradicts the conditions for identifying a parabola.

Explanation
The equation x = -cos(t) represents a horizontal line that moves from -1 to 1 as t varies from 0 to 180. The equation y = sin^2(t) represents a curve that oscillates between 0 and 1 as t varies from 0 to 180. When we plot these points on a graph, we can see that they form a parabolic shape. Therefore, the conic represented by the given equation is a parabola.

Explanation
After drawing a parametric graph, we need to draw arrows using the t values. This is because the t values represent the parameter that determines the position of the point on the graph. By drawing arrows along the graph using the t values, we can indicate the direction in which the parameter is increasing or decreasing. This helps in understanding the movement and behavior of the graph.