Find equations of the osculating circles of the ellipse 9x^2+4y^2=36 at the points (2,0)&(0,3).
To find the equations of the osculating circles of the ellipse at the given points, we need to find the center and radius of each osculating circle.
The equation of an ellipse centered at the origin is given by:
(x^2/a^2) + (y^2/b^2) = 1
Comparing this with the given equation (9x^2 + 4y^2 = 36), we see that a^2 = 36/9 = 4 and b^2 = 36/4 = 9.
Now, the equation of the osculating circle at any given point (x0, y0) on the ellipse can be written as:
(x - x0)^2 + (y - y0)^2 = r^2
where (x0, y0) is the center of the osculating circle and r is its radius.
To find the center of the osculating circle, we need to find the first and second derivatives of the given equation and evaluate them at each point.
First, let's differentiate the given equation with respect to x:
18x + 8yy' = 0 ...(1)
where y' denotes dy/dx (the first derivative of y with respect to x).
Similarly, let's differentiate the given equation with respect to y:
8xx' + 8y = 0 ...(2)
where x' denotes dx/dy (the first derivative of x with respect to y).
Now, let's find the values of y' (dy/dx) and x' (dx/dy) at each point.
At (2, 0):
Substituting x = 2 into equation (1), we get:
36 + 8(0)y' = 0
y' = 0
Substituting y = 0 into equation (2), we get:
8(2)x' = 0
x' = 0
So, at (2, 0), y' = 0 and x' = 0.
At (0, 3):
Substituting x = 0 into equation (1), we get:
0 + 8(3)y' = 0
y' = 0
Substituting y = 3 into equation (2), we get:
8(0)x' + 24 = 0
24 = 0 [This is not possible]
At (0, 3), y' = 0 and we cannot find x'.
Now, let's find the radius (r) of each osculating circle using the formula:
r = abs(y'x - x'y) / sqrt((x')^2 + (y')^2)
At (2, 0):
Substituting x = 2 and y = 0 into the formula, we get:
r = abs(0(2) - 0(0)) / sqrt((0)^2 + (0)^2)
r = 0
At (0, 3):
Substituting x = 0 and y = 3 into the formula, we get:
r = abs(0(0) - 0(3)) / sqrt((0)^2 + (0)^2)
r = 0
Therefore, at both points (2, 0) and (0, 3), the osculating circles degenerate into a point.
Thus, the equations of the osculating circles at the points (2, 0) and (0, 3) are x^2 + y^2 = 0.
This is just more of the same.
Find r=1/k. Then you have the radius of the circle, and you can easily find its center.
Then just use the standard form for a circle.