Part 3/6:
On the other hand, consider the family of straight lines defined by the equation ( y = k \cdot x ); these lines also change based on the constant ( k ). By plotting these curves together, you can assess whether the lines and circles are indeed orthogonal.
Proving Orthogonality
To demonstrate that two families of curves are orthogonal, we need to utilize implicit differentiation to find the derivatives of each curve. The crux of proving orthogonality lies in showing that the slopes (derivatives) of the curves at the points of intersection are negative reciprocals of each other.
Calculating Derivatives
For the circle defined by ( x^2 + y^2 = R^2 ):
- Taking the derivative implicitly using the power rule gives:
[ 2x + 2y \frac{dy}{dx} = 0 ]
- Rearranging yields: