Find the orthogonal trajectory of the family of circle x^2 (y-c)^2 c^2 where c is parameter.
it is a family of lines through the center. See what you can do with that.
To find the orthogonal trajectory of the given family of circles, we first need to find the differential equation satisfied by the family of circles and then solve it.
The equation of a circle with center (0, c) and radius c is given by:
x^2 + (y - c)^2 = c^2
Differentiating both sides with respect to x, we get:
2x + 2(y - c)(dy/dx) = 0
(y - c)(dy/dx) = -x
We can rewrite this in a more standard form by solving for dy/dx:
dy/dx = -x / (y - c)
Now, to find the orthogonal trajectory, we need to find the negative reciprocal of dy/dx. Let's call it m.
m = (y - c) / x
To eliminate the constant c, we differentiate the above equation with respect to x:
dm/dx = [(dy/dx)(x) - (y - c)] / x^2
dm/dx = (-(x^2 / (y - c)) - (y - c)) / x^2
dm/dx = -(x^2 + (y - c)^2) / (x^2(y - c))
This equation represents the differential equation of the orthogonal trajectory.
Now, let's solve this differential equation to find the orthogonal trajectory.
(dm/dx) * (x^2(y - c)) = -(x^2 + (y - c)^2)
Expanding and simplifying, we get:
-x^2(y - c) - (y - c)^2 = 0
-(x^2(y - c) + (y - c)^2) = 0
-(y - c)(x^2 + (y - c)) = 0
This equation represents the orthogonal trajectory.
So, the orthogonal trajectory of the given family of circles is given by the equation:
-(y - c)(x^2 + (y - c)) = 0
To find the orthogonal trajectory of a family of curves, we need to find a curve that intersects all curves in the family at right angles.
Let's start by finding the differential equation representing the family of circles. The given family of circles has the equation:
x^2 + (y - c)^2 = c^2
To find the differential equation, we'll differentiate both sides of the equation with respect to x. Assuming that y = y(x) and c = c(x), we have:
2x + 2(y - c)(dy/dx) = 2c(dc/dx)
Simplifying the equation, we get:
x + (y - c)(dy/dx) = c(dc/dx)
We can rewrite this equation into a more standard form:
dy/dx = (c - y)/(x - c)
Now, to find the orthogonal trajectory, we need to find the curve that is perpendicular to the family of circles. We know that the product of the slopes of two perpendicular curves is -1. So, we'll take the negative reciprocal of the slope of the family of circles to find the slope of the orthogonal trajectory.
Recall that the slope of the family of circles is (c - y)/(x - c). Therefore, the slope of the orthogonal trajectory is:
-1/(c - y)/(x - c) = (x - c)/(y - c)
Now, we have the differential equation representing the orthogonal trajectory:
dy/dx = (x - c)/(y - c)
To find the curve that satisfies this differential equation, we can solve it using standard differential equation techniques, such as separation of variables or integrating factors.
I hope this helps you understand how to find the orthogonal trajectory of a family of curves.