The graph of the equation (x-h)^2/(a^2) + (y-k)^2/b^2 = 1 is an ellipse with center (h,k), horizontal axis length 2a, and vertical axis length 2b. Find parametric equations whose graph is an ellipse with center (h,k), horizontal axis length 2a, and vertical axis length 2b, and explain why your answer is correct.
I really have no idea where to begin. I'm sorta new to the concept of parametric equations, so could someone please explain how to solve this? Thanks!
you know that
x=r cosθ
y=r sinθ
describes a circle of radius r, with center at (0,0).
If the axes are a and b, instead of r,
x = a cosθ
y = b sinθ
If the center is at (h,k) just translate the coordinates:
x = a cosθ + h
y = b sinθ + k
(x-h)^2 /a^2 + (y-k)^2 /b^2 = 1
we know that sin^2 t + cos^2 t = 1
so let cos^2 t = (x-h)^2/a^2
cos t = (x-h)/a
and let sin^2 t = (y-k)^2 / b^2
sin t = (y-k)/b
from
(x-h)/a = cos t
x-h = acos t
x = h + a cos t
from
(y-k)/b = sin t
y-k = b sin t
y = k + b sint
thus:
x=h + acost
y = k + bsint
Here is a Youtube that might shed light on this
http://www.youtube.com/watch?v=zs0Nw0tb4y8
Thank you very much both of you sirs it is much appreciated!
To find parametric equations for the given ellipse, we can use the parameterization method. The general idea is to create two equations, one for x and one for y, in terms of a parameter, typically denoted by t.
1. First, let's rewrite the equation of the ellipse in standard form:
(x - h)^2 / a^2 + (y - k)^2 / b^2 = 1
2. We can isolate y in terms of x from the equation above. Start by multiplying both sides by b^2:
(x - h)^2 / a^2 + (y - k)^2 = b^2
Multiply through by a^2:
a^2(x - h)^2 + b^2(y - k)^2 = a^2b^2
Expand:
a^2x^2 - 2a^2hx + a^2h^2 + b^2y^2 - 2b^2ky + b^2k^2 = a^2b^2
Rearrange:
y = (sqrt(a^2b^2 - b^2(x - h)^2) + bk) / b^2 + k,
y = (-sqrt(a^2b^2 - b^2(x - h)^2) + bk) / b^2 + k
3. From the equations derived in step 2, we can see that y depends on x, h, k, a, and b. To write parametric equations, we will set a parameter, t, which represents a point along the ellipse.
4. We'll set x = h + a * cos(t) as the parametric equation for x. This equation represents the x-coordinate of points on an ellipse centered at (h, k) with a horizontal axis length of 2a. The parameter t is used to control the position of the point along the ellipse, ranging from 0 to 2π (or any desired range for a complete ellipse).
5. Now substitute x = h + a * cos(t) into the equations derived in step 2 to get the parametric equations for y:
y = (sqrt(a^2b^2 - b^2(h + a * cos(t) - h)^2) + bk) / b^2 + k
y = (-sqrt(a^2b^2 - b^2(h + a * cos(t) - h)^2) + bk) / b^2 + k
6. Simplifying further:
y = (sqrt(a^2b^2 - b^2a^2 * cos^2(t)) + bk) / b^2 + k
y = (-sqrt(a^2b^2 - b^2a^2 * cos^2(t)) + bk) / b^2 + k
By using these parametric equations, the values of x and y will vary simultaneously as t changes, effectively tracing the graph of the ellipse centered at (h, k) with horizontal axis length 2a and vertical axis length 2b.
Explanation of correctness:
The parametric equations x = h + a * cos(t) and y = ... are derived from the original equation of the ellipse. By substituting these equations into the original equation, we obtain equations that satisfy the given conditions of the ellipse, including the center, axis lengths, and shape. Furthermore, the parameter t governs the position along the ellipse, allowing us to trace out all the points on the ellipse by varying t. Therefore, the parametric equations are correct representations of points on the ellipse.